Theorem 3anibar

Description: Remove a hypothesis from the second member of a biconditional. (Contributed by FL, 22-Jul-2008)

	Ref	Expression
Hypothesis	3anibar.1	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \left(\theta \leftrightarrow \left(\chi \wedge \tau \right)\right)$$
Assertion	3anibar	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \left(\theta \leftrightarrow \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		3anibar.1	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \left(\theta \leftrightarrow \left(\chi \wedge \tau \right)\right)$$
2		simp3	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \chi$$
3	2 1	mpbirand	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \left(\theta \leftrightarrow \tau \right)$$