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  947  isset  2807  rexcom4b  2826  eueq  2975  ssrabeq  3312  a9evsep  4209  pwunim  4381  elvv  4786  elvvv  4787  resopab  5055  funfn  5354  dffn2  5481  dffn3  5490  dffn4  5562  fsn  5815  ixp0x  6890  ac6sfi  7080  fimax2gtri  7084  nninfwlporlemd  7362  ccatrcan  11290  xrmaxiflemcom  11800  plyun0  15450  trirec0xor  16585