Theorem biorf 663

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 632	. 2 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
2		orel1 644	. 2 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
3	1, 2	impbid2 131	1 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629

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 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  biortn  664  pm5.61  708  pm5.55dc  819  euor  1926  eueq3dc  2715  difprsnss  3502  opthprc  4391  frecsuclem3  5990  swoord1  6135  indpi  6440  enq0tr  6532  mulap0r  7606  mulge0  7610  leltap  7615  ap0gt0  7629