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  3600  difprsnss  3760  exmidsssn  4235  opthprc  4714  frecabcl  6457  frecsuclem  6464  swoord1  6621  indpi  7409  enq0tr  7501  mulap0r  8642  mulge0  8646  leltap  8652  ap0gt0  8667  sumsplitdc  11597  coprm  12312  gsumval2  13040  bdbl  14739  subctctexmid  15645