Theorem biorf 751

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 718	. 2 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
2		orel1 732	. 2 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
3	1, 2	impbid2 143	1 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 715

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 716

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biortn  752  pm5.61  801  pm5.55dc  920  3bior1fd  1388  3bior2fd  1390  euor  2105  eueq3dc  2980  ifordc  3647  difprsnss  3811  exmidsssn  4292  opthprc  4777  frecabcl  6564  frecsuclem  6571  swoord1  6730  indpi  7561  enq0tr  7653  mulap0r  8794  mulge0  8798  leltap  8804  ap0gt0  8819  sumsplitdc  11992  coprm  12715  gsumval2  13479  bdbl  15226  eupth2lem1  16308  eupth2lem2dc  16309  subctctexmid  16601