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  949  isset  2809  rexcom4b  2828  eueq  2977  ssrabeq  3314  a9evsep  4211  pwunim  4383  elvv  4788  elvvv  4789  resopab  5057  funfn  5356  dffn2  5484  dffn3  5493  dffn4  5565  fsn  5819  ixp0x  6894  ac6sfi  7086  fimax2gtri  7090  nninfwlporlemd  7370  ccatrcan  11299  xrmaxiflemcom  11809  plyun0  15459  trirec0xor  16649