Theorem biorf 394

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 373	. 2 ⊢ (ψ → (φ ∨ ψ))
2		orel1 371	. 2 ⊢ (¬ φ → ((φ ∨ ψ) → ψ))
3	1, 2	impbid2 195	1 ⊢ (¬ φ → (ψ ↔ (φ ∨ ψ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359

This theorem is referenced by:  biortn  395  pm5.61  693  pm5.55  867  cadan  1392  euor  2231  eueq3  3012  unineq  3506  ifor  3703  difprsnss  3847  eqtfinrelk  4487  dfphi2  4570