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Mirrors > Home > MPE Home > Th. List > neor | Structured version Visualization version GIF version |
Description: Logical OR with an equality. (Contributed by NM, 29-Apr-2007.) |
Ref | Expression |
---|---|
neor | ⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (𝐴 ≠ 𝐵 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-or 384 | . 2 ⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (¬ 𝐴 = 𝐵 → 𝜓)) | |
2 | df-ne 2782 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 2 | imbi1i 338 | . 2 ⊢ ((𝐴 ≠ 𝐵 → 𝜓) ↔ (¬ 𝐴 = 𝐵 → 𝜓)) |
4 | 1, 3 | bitr4i 266 | 1 ⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (𝐴 ≠ 𝐵 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 = wceq 1475 ≠ wne 2780 |
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 196 df-or 384 df-ne 2782 |
This theorem is referenced by: frsn 5112 ord0eln0 5696 fimaxre 10847 prime 11334 h1datomi 27824 elat2 28583 bnj563 30067 divrngidl 32997 dmncan1 33045 lkrshp4 33413 cvrcmp 33588 leat2 33599 isat3 33612 2llnmat 33828 2lnat 34088 |
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