Theorem biorfi 697

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 665	. . 3 ⊢ (𝜓 → (𝜓 ∨ 𝜑))
3		orel2 677	. . 3 ⊢ (¬ 𝜑 → ((𝜓 ∨ 𝜑) → 𝜓))
4	2, 3	impbid2 141	. 2 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜓 ∨ 𝜑)))
5	1, 4	ax-mp 7	1 ⊢ (𝜓 ↔ (𝜓 ∨ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 103   ∨ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm4.43  890  dn1dc  901  excxor  1309  un0  3278  opthprc  4409  frec0g  6006  dcdc  10572