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  2777  rexcom4b  2796  eueq  2943  ssrabeq  3279  a9evsep  4165  pwunim  4332  elvv  4736  elvvv  4737  resopab  5002  funfn  5300  dffn2  5426  dffn3  5435  dffn4  5503  fsn  5751  ixp0x  6812  ac6sfi  6994  fimax2gtri  6997  nninfwlporlemd  7273  xrmaxiflemcom  11531  plyun0  15179  trirec0xor  15946