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  393  mpbiran2  954  isset  2828  rexcom4b  2847  eueq  2997  ssrabeq  3336  a9evsep  4250  pwunim  4426  elvv  4832  elvvv  4833  resopab  5102  funfn  5402  dffn2  5530  dffn3  5539  dffn4  5616  fsn  5871  ixp0x  6998  ac6sfi  7192  fimax2gtri  7196  nninfwlporlemd  7502  ccatrcan  11469  xrmaxiflemcom  11993  plyun0  15760  trirec0xor  16999