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  4181  regexmidlem1  4625  peano5  4690  intirr  5115  fvopab6  5733  riotav  5966  mpov  6100  brtpos0  6404  frec0g  6549  inl11  7243  apreim  8761  ccatlcan  11265  clim0  11811  gcd0id  12515  nnwosdc  12575  gsum0g  13444  isbasis3g  14735  opnssneib  14845  ssidcn  14899  bj-d0clsepcl  16343