Theorem bian1d

Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017)

	Ref	Expression
Hypothesis	bian1d.1	|- ( ph -> ( ps <-> ( ch /\ th ) ) )
Assertion	bian1d	|- ( ph -> ( ( ch /\ ps ) <-> ( ch /\ th ) ) )

Proof

Step	Hyp	Ref	Expression
1		bian1d.1	|- ( ph -> ( ps <-> ( ch /\ th ) ) )
2	1	biimpd	|- ( ph -> ( ps -> ( ch /\ th ) ) )
3	2	adantld	|- ( ph -> ( ( ch /\ ps ) -> ( ch /\ th ) ) )
4		simpl	|- ( ( ch /\ th ) -> ch )
5	4	a1i	|- ( ph -> ( ( ch /\ th ) -> ch ) )
6	1	biimprd	|- ( ph -> ( ( ch /\ th ) -> ps ) )
7	5 6	jcad	|- ( ph -> ( ( ch /\ th ) -> ( ch /\ ps ) ) )
8	3 7	impbid	|- ( ph -> ( ( ch /\ ps ) <-> ( ch /\ th ) ) )