Biconditional Elimination/Proof Rule

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Theorem

The Rule of Biconditional Elimination is a valid deduction sequent in propositional logic.

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:

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


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


Tableau Form

Let $\phi \iff \psi$ be a propositional 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 $\iff \mathcal E_1$             


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 $\iff \mathcal E_2$             


Also known as

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


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