Theorem neor 3035

Description: Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)

Assertion

Ref	Expression
neor	⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (𝐴 ≠ 𝐵 → 𝜓))

Proof of Theorem neor

Step	Hyp	Ref	Expression
1		df-or 844	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (¬ 𝐴 = 𝐵 → 𝜓))
2		df-ne 2943	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	imbi1i 349	. 2 ⊢ ((𝐴 ≠ 𝐵 → 𝜓) ↔ (¬ 𝐴 = 𝐵 → 𝜓))
4	1, 3	bitr4i 277	1 ⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (𝐴 ≠ 𝐵 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 843   = wceq 1539   ≠ wne 2942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-or 844  df-ne 2943

This theorem is referenced by:  frsn  5665  ord0eln0  6305  fimaxre  11849  fiminre  11852  prime  12331  h1datomi  29844  elat2  30603  bnj563  32623  divrngidl  36113  dmncan1  36161  lkrshp4  37049  cvrcmp  37224  leat2  37235  isat3  37248  2llnmat  37465  2lnat  37725