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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 846 | . 2 ⊢ ((𝐴 = 𝐵 ∨ 𝜓) ↔ (¬ 𝐴 = 𝐵 → 𝜓)) | |
2 | df-ne 2941 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1541 ≠ wne 2940 |
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 846 df-ne 2941 |
This theorem is referenced by: frsn 5763 ord0eln0 6419 fimaxre 12157 fiminre 12160 prime 12642 h1datomi 30829 elat2 31588 bnj563 33749 divrngidl 36891 dmncan1 36939 lkrshp4 37973 cvrcmp 38148 leat2 38159 isat3 38172 2llnmat 38390 2lnat 38650 |
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