Biconditional Elimination

Theorem
The rule of biconditional elimination is a valid deduction sequent in propositional logic:

This is two proof rules in one:
 * $(1): \quad$ If we can conclude $p \iff q$, then we may infer $p \implies q$.
 * $(2): \quad$ If we can conclude $p \iff q$, then we may infer $q \implies p$.

It can be written:
 * $\displaystyle {p \iff q \over p \implies q} \iff_{e_1} \qquad \qquad {p \iff q \over q \implies p} \iff_{e_2}$

Explanation
Note that here there are two proof rules in one. The first of the two tells us that, given a biconditional, we may infer a conditional from the first to the second of its constituents, while the second says that, given a biconditional, we may infer a conditional from the second to the first of its constituents.

At this stage, such attention to detail is important.

Technical Note
When invoking Biconditional Elimination in a tableau proof, use the BiconditionalElimination template:



or:

where:
 * is the number of the line on the tableau proof where the Biconditional Elimination is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the line of the tableau proof upon which this line directly depends
 * should hold 1 for, and 2 for
 * is the (optional) comment that is to be displayed in the Notes column.