Theorem mpbiran 942

Description: Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
mpbiran	|- ( ph <-> ch )

Proof of Theorem mpbiran

Step	Hyp	Ref	Expression
1		mpbiran.2	. 2 |- ( ph <-> ( ps /\ ch ) )
2		mpbiran.1	. . 3 |- ps
3	2	biantrur 303	. 2 |- ( ch <-> ( ps /\ ch ) )
4	1, 3	bitr4i 187	1 |- ( ph <-> ch )

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:  mpbir2an  944  unssdif  3407  unssin  3411  inssun  3412  invdif  3414  pwpwab  4014  exmidexmid  4239  opabm  4326  regexmidlem1  4580  elirr  4588  en2lp  4601  wessep  4625  peano5  4645  relop  4827  ssrnres  5124  funopab  5305  funcnv2  5333  funcnveq  5336  fnres  5391  idref  5824  rnoprab  6027  elixp  6791  djuf1olem  7154  lbfzo0  10303  expghmap  14311  txdis1cn  14692