Theorem biantrur 303

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

Hypothesis

Ref	Expression
biantrur.1	⊢ 𝜑

Assertion

Ref	Expression
biantrur	⊢ (𝜓 ↔ (𝜑 ∧ 𝜓))

Proof of Theorem biantrur

Step	Hyp	Ref	Expression
1		biantrur.1	. 2 ⊢ 𝜑
2		ibar 301	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  mpbiran  953  truan  1419  rexv  2840  reuv  2841  rmov  2842  rabab  2843  euxfrdc  3012  euind  3013  dfdif3  3339  ddifstab  3361  vss  3568  mptv  4230  regexmidlem1  4682  peano5  4747  intirr  5177  fvopab6  5808  riotav  6047  mpov  6181  opabn1stprc  6433  brtpos0  6527  frec0g  6672  inl11  7411  apreim  8939  ccatlcan  11512  clim0  12076  gcd0id  12781  nnwosdc  12841  gzsum0  13772  isbasis3g  15272  opnssneib  15382  ssidcn  15436  bj-d0clsepcl  17156