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  2068  eueq3dc  2934  ifordc  3596  difprsnss  3756  exmidsssn  4231  opthprc  4710  frecabcl  6452  frecsuclem  6459  swoord1  6616  indpi  7402  enq0tr  7494  mulap0r  8634  mulge0  8638  leltap  8644  ap0gt0  8659  sumsplitdc  11575  coprm  12282  gsumval2  12980  bdbl  14671  subctctexmid  15491