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  946  truan  1412  rexv  2818  reuv  2819  rmov  2820  rabab  2821  euxfrdc  2989  euind  2990  dfdif3  3314  ddifstab  3336  vss  3539  mptv  4181  regexmidlem1  4625  peano5  4690  intirr  5115  fvopab6  5733  riotav  5966  mpov  6100  brtpos0  6404  frec0g  6549  inl11  7243  apreim  8761  ccatlcan  11266  clim0  11812  gcd0id  12516  nnwosdc  12576  gsum0g  13445  isbasis3g  14736  opnssneib  14846  ssidcn  14900  bj-d0clsepcl  16371