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  2806  rexcom4b  2825  eueq  2974  ssrabeq  3311  a9evsep  4206  pwunim  4377  elvv  4781  elvvv  4782  resopab  5049  funfn  5348  dffn2  5475  dffn3  5484  dffn4  5556  fsn  5809  ixp0x  6881  ac6sfi  7068  fimax2gtri  7072  nninfwlporlemd  7350  ccatrcan  11266  xrmaxiflemcom  11775  plyun0  15425  trirec0xor  16473