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  943  isset  2769  rexcom4b  2788  eueq  2935  ssrabeq  3270  a9evsep  4155  pwunim  4321  elvv  4725  elvvv  4726  resopab  4990  funfn  5288  dffn2  5409  dffn3  5418  dffn4  5486  fsn  5734  ixp0x  6785  ac6sfi  6959  fimax2gtri  6962  nninfwlporlemd  7238  xrmaxiflemcom  11414  plyun0  14972  trirec0xor  15689