Biconditional Elimination/Proof Rule

Theorem

The rule of biconditional elimination is a valid argument in types of logic dealing with conditionals $\implies$ and biconditionals $\iff$.

This includes classical propositional logic and predicate logic, and in particular natural deduction.

As a proof rule it is expressed in either of the two forms:

$(1): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\phi \implies \psi$.

$(2): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\psi \implies \phi$.

It can be written:

$\ds {\phi \iff \psi \over \phi \implies \psi} {\iff}_{e_1} \qquad \text{or} \qquad {\phi \iff \psi \over \psi \implies \phi} {\iff}_{e_2}$

Thus it is used to introduce the biconditional operator into a sequent.

Tableau Form

Let $\phi \iff \psi$ be a well-formed formula] in a tableau proof whose main connective is the biconditional operator.

Biconditional Elimination is invoked for $\phi \iff \psi$ in either of the two forms:

Form 1

			Pool:					The pooled assumptions of $\phi \iff \psi$
			Formula:					$\phi \implies \psi$
			Description:					Biconditional Elimination
			Depends on:					The line containing $\phi \iff \psi$
			Abbreviation:					$\mathrm {BE}_1$ or $\iff \EE_1$

Form 2

			Pool:					The pooled assumptions of $\phi \iff \psi$
			Formula:					$\psi \implies \phi$
			Description:					Biconditional Elimination
			Depends on:					The line containing $\phi \iff \psi$
			Abbreviation:					$\mathrm {BE}_2$ or $\iff \EE_2$

Also known as

Some sources refer to the Biconditional Elimination as the rule of Biconditional-Conditional.

Also see

• This is a rule of inference of the following proof systems:
  • Definition:Natural Deduction

Technical Note

When invoking Biconditional Elimination in a tableau proof, use the {{BiconditionalElimination}} template:

{{BiconditionalElimination|line|pool|statement|depend|1 or 2}}

or:

{{BiconditionalElimination|line|pool|statement|depend|1 or 2|comment}}

where:

line is the number of the line on the tableau proof where Biconditional Elimination is to be invoked

pool is the pool of assumptions (comma-separated list)

statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters

depend is the line of the tableau proof upon which this line directly depends

1 or 2 should hold 1 for BiconditionalElimination_1, and 2 for BiconditionalElimination_2

comment is the (optional) comment that is to be displayed in the Notes column.

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction: Exercises $1.6: \ 7$