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  4205  pwunim  4376  elvv  4780  elvvv  4781  resopab  5048  funfn  5347  dffn2  5474  dffn3  5483  dffn4  5553  fsn  5806  ixp0x  6871  ac6sfi  7056  fimax2gtri  7059  nninfwlporlemd  7335  ccatrcan  11246  xrmaxiflemcom  11755  plyun0  15404  trirec0xor  16372