Theorem biorf 745

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 712	. 2 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
2		orel1 726	. 2 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
3	1, 2	impbid2 143	1 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biortn  746  pm5.61  795  pm5.55dc  914  euor  2071  eueq3dc  2938  ifordc  3601  difprsnss  3761  exmidsssn  4236  opthprc  4715  frecabcl  6466  frecsuclem  6473  swoord1  6630  indpi  7428  enq0tr  7520  mulap0r  8661  mulge0  8665  leltap  8671  ap0gt0  8686  sumsplitdc  11616  coprm  12339  gsumval2  13101  bdbl  14847  subctctexmid  15755