Theorem neor 3106

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 3015	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	imbi1i 352	. 2 ⊢ ((𝐴 ≠ 𝐵 → 𝜓) ↔ (¬ 𝐴 = 𝐵 → 𝜓))
4	1, 3	bitr4i 280	1 ⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (𝐴 ≠ 𝐵 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843   = wceq 1531   ≠ wne 3014

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 209  df-or 844  df-ne 3015

This theorem is referenced by:  frsn  5632  ord0eln0  6238  fimaxre  11576  fimaxreOLD  11577  fiminre  11580  prime  12055  h1datomi  29350  elat2  30109  bnj563  32007  divrngidl  35298  dmncan1  35346  lkrshp4  36236  cvrcmp  36411  leat2  36422  isat3  36435  2llnmat  36652  2lnat  36912