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))

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  4180  regexmidlem1  4624  peano5  4689  intirr  5114  fvopab6  5730  riotav  5959  mpov  6093  brtpos0  6396  frec0g  6541  inl11  7228  apreim  8746  ccatlcan  11245  clim0  11791  gcd0id  12495  nnwosdc  12555  gsum0g  13424  isbasis3g  14714  opnssneib  14824  ssidcn  14878  bj-d0clsepcl  16246