Theorem biorfi 736

Description: A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)

Hypothesis

Ref	Expression
biorfi.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
biorfi	⊢ (𝜓 ↔ (𝜓 ∨ 𝜑))

Proof of Theorem biorfi

Step	Hyp	Ref	Expression
1		biorfi.1	. 2 ⊢ ¬ 𝜑
2		orc 702	. . 3 ⊢ (𝜓 → (𝜓 ∨ 𝜑))
3		orel2 716	. . 3 ⊢ (¬ 𝜑 → ((𝜓 ∨ 𝜑) → 𝜓))
4	2, 3	impbid2 142	. 2 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜓 ∨ 𝜑)))
5	1, 4	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜓 ∨ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ wo 698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm4.43  939  dn1dc  950  excxor  1368  un0  3442  opthprc  4655  frec0g  6365  if0ab  13697