Theorem bian1d 29306

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

Hypothesis

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

Assertion

Ref	Expression
bian1d	|- ( ph -> ( ( ch /\ ps ) <-> ( ch /\ th ) ) )

Proof of Theorem bian1d

Step	Hyp	Ref	Expression
1		bian1d.1	. . . 4 |- ( ph -> ( ps <-> ( ch /\ th ) ) )
2	1	biimpd 219	. . 3 |- ( ph -> ( ps -> ( ch /\ th ) ) )
3	2	adantld 483	. 2 |- ( ph -> ( ( ch /\ ps ) -> ( ch /\ th ) ) )
4		simpl 473	. . . 4 |- ( ( ch /\ th ) -> ch )
5	4	a1i 11	. . 3 |- ( ph -> ( ( ch /\ th ) -> ch ) )
6	1	biimprd 238	. . 3 |- ( ph -> ( ( ch /\ th ) -> ps ) )
7	5, 6	jcad 555	. 2 |- ( ph -> ( ( ch /\ th ) -> ( ch /\ ps ) ) )
8	3, 7	impbid 202	1 |- ( ph -> ( ( ch /\ ps ) <-> ( ch /\ th ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  funcnvmptOLD  29467  funcnvmpt  29468