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  944  isset  2778  rexcom4b  2797  eueq  2944  ssrabeq  3280  a9evsep  4166  pwunim  4333  elvv  4737  elvvv  4738  resopab  5003  funfn  5301  dffn2  5427  dffn3  5436  dffn4  5504  fsn  5752  ixp0x  6813  ac6sfi  6995  fimax2gtri  6998  nninfwlporlemd  7274  xrmaxiflemcom  11560  plyun0  15208  trirec0xor  15984