Theorem biorfi 664

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 632	. . 3 ⊢ (ψ → (ψ ∨ φ))
3		orel2 644	. . 3 ⊢ (¬ φ → ((ψ ∨ φ) → ψ))
4	2, 3	impbid2 131	. 2 ⊢ (¬ φ → (ψ ↔ (ψ ∨ φ)))
5	1, 4	ax-mp 7	1 ⊢ (ψ ↔ (ψ ∨ φ))

Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 628

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm4.43  855  dn1dc  866  excxor  1268  un0  3245  opthprc  4334  frec0g  5922  dcdc  9236