Theorem biantru 302

Description: A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
biantru.1	⊢ 𝜑

Assertion

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

Proof of Theorem biantru

Step	Hyp	Ref	Expression
1		biantru.1	. 2 ⊢ 𝜑
2		iba 300	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜓 ∧ 𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm4.71  389  mpbiran2  950  isset  2810  rexcom4b  2829  eueq  2978  ssrabeq  3316  a9evsep  4216  pwunim  4389  elvv  4794  elvvv  4795  resopab  5063  funfn  5363  dffn2  5491  dffn3  5500  dffn4  5574  fsn  5827  ixp0x  6938  ac6sfi  7130  fimax2gtri  7134  nninfwlporlemd  7414  ccatrcan  11349  xrmaxiflemcom  11872  plyun0  15530  trirec0xor  16760