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  5731  riotav  5960  mpov  6094  brtpos0  6398  frec0g  6543  inl11  7232  apreim  8750  ccatlcan  11250  clim0  11796  gcd0id  12500  nnwosdc  12560  gsum0g  13429  isbasis3g  14720  opnssneib  14830  ssidcn  14884  bj-d0clsepcl  16288