Theorem biorf 662

Description: A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)

Assertion

Ref	Expression
biorf	⊢ (¬ φ → (ψ ↔ (φ ∨ ψ)))

Proof of Theorem biorf

Step	Hyp	Ref	Expression
1		olc 631	. 2 ⊢ (ψ → (φ ∨ ψ))
2		orel1 643	. 2 ⊢ (¬ φ → ((φ ∨ ψ) → ψ))
3	1, 2	impbid2 131	1 ⊢ (¬ φ → (ψ ↔ (φ ∨ ψ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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:  biortn  663  pm5.61  707  pm5.55dc  818  euor  1923  eueq3dc  2709  difprsnss  3493  opthprc  4334  frecsuclem3  5929  swoord1  6071  indpi  6326  enq0tr  6416  mulap0r  7359  mulge0  7363