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  3271  a9evsep  4156  pwunim  4322  elvv  4726  elvvv  4727  resopab  4991  funfn  5289  dffn2  5412  dffn3  5421  dffn4  5489  fsn  5737  ixp0x  6794  ac6sfi  6968  fimax2gtri  6971  nninfwlporlemd  7247  xrmaxiflemcom  11431  plyun0  15056  trirec0xor  15776