Theorem biantrur 303

Description: A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)

Hypothesis

Ref	Expression
biantrur.1	|- ph

Assertion

Ref	Expression
biantrur	|- ( ps <-> ( ph /\ ps ) )

Proof of Theorem biantrur

Step	Hyp	Ref	Expression
1		biantrur.1	. 2 |- ph
2		ibar 301	. 2 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
3	1, 2	ax-mp 5	1 |- ( ps <-> ( ph /\ ps ) )

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

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

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mpbiran  953  truan  1419  rexv  2840  reuv  2841  rmov  2842  rabab  2843  euxfrdc  3012  euind  3013  dfdif3  3339  ddifstab  3361  vss  3568  mptv  4226  regexmidlem1  4678  peano5  4743  intirr  5172  fvopab6  5799  riotav  6038  mpov  6172  opabn1stprc  6423  brtpos0  6517  frec0g  6662  inl11  7399  apreim  8925  ccatlcan  11473  clim0  12034  gcd0id  12739  nnwosdc  12799  gzsum0  13696  isbasis3g  15130  opnssneib  15240  ssidcn  15294  bj-d0clsepcl  16934