Theorem mpbiran2 949

Description: Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)

Hypotheses

Ref	Expression
mpbiran2.1	|- ch
mpbiran2.2	|- ( ph <-> ( ps /\ ch ) )

Assertion

Ref	Expression
mpbiran2	|- ( ph <-> ps )

Proof of Theorem mpbiran2

Step	Hyp	Ref	Expression
1		mpbiran2.2	. 2 |- ( ph <-> ( ps /\ ch ) )
2		mpbiran2.1	. . 3 |- ch
3	2	biantru 302	. 2 |- ( ps <-> ( ps /\ ch ) )
4	1, 3	bitr4i 187	1 |- ( ph <-> ps )

Syntax hints:   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  reueq  3005  ss0b  3534  eusv1  4549  eusv2nf  4553  eusv2  4554  opthprc  4777  opelres  5018  f1cnvcnv  5553  fores  5569  f1orn  5593  funfvdm  5709  dfoprab2  6067  tpostpos  6429  opelreal  8046  elreal2  8049  eqresr  8055  axprecex  8099  zeoxor  12429  isprm2  12688  toptopon  14741  bdeq0  16462  subctctexmid  16601