Theorem biorf 752

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 719	. 2 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
2		orel1 733	. 2 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
3	1, 2	impbid2 143	1 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 716

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 620  ax-io 717

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biortn  753  pm5.61  802  pm5.55dc  921  3bior1fd  1389  3bior2fd  1391  euor  2108  eueq3dc  2994  ifordc  3669  difprsnss  3838  exmidsssn  4321  opthprc  4807  frecabcl  6644  frecsuclem  6651  swoord1  6810  indpi  7674  enq0tr  7766  mulap0r  8908  mulge0  8912  leltap  8918  ap0gt0  8933  sumsplitdc  12148  coprm  12871  gsumval2  13665  bdbl  15499  eupth2lem1  16584  eupth2lem2dc  16585  eupth2lem3lem6fi  16597  subctctexmid  16915