Theorem bimsc1

Description: Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	bimsc1	$$⊢\left(\left(\phi \to \psi \right)\wedge \left(\chi \leftrightarrow \left(\psi \wedge \phi \right)\right)\right)\to \left(\chi \leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\left(\chi \leftrightarrow \left(\psi \wedge \phi \right)\right)\to \left(\chi \leftrightarrow \left(\psi \wedge \phi \right)\right)$$
2		simpr	$$⊢\left(\psi \wedge \phi \right)\to \phi$$
3		ancr	$$⊢\left(\phi \to \psi \right)\to \left(\phi \to \left(\psi \wedge \phi \right)\right)$$
4	2 3	impbid2	$$⊢\left(\phi \to \psi \right)\to \left(\left(\psi \wedge \phi \right)\leftrightarrow \phi \right)$$
5	1 4	sylan9bbr	$$⊢\left(\left(\phi \to \psi \right)\wedge \left(\chi \leftrightarrow \left(\psi \wedge \phi \right)\right)\right)\to \left(\chi \leftrightarrow \phi \right)$$