| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brneqtrd | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
| Ref | Expression |
|---|---|
| brneqtrd.1 | ⊢ (𝜑 → ¬ 𝐴𝑅𝐵) |
| brneqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| brneqtrd | ⊢ (𝜑 → ¬ 𝐴𝑅𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brneqtrd.1 | . 2 ⊢ (𝜑 → ¬ 𝐴𝑅𝐵) | |
| 2 | brneqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
| 3 | 2 | breq2d 5135 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)) |
| 4 | 1, 3 | mtbid 324 | 1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 class class class wbr 5123 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2706 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2713 df-cleq 2726 df-clel 2808 df-rab 3420 df-v 3465 df-dif 3934 df-un 3936 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-br 5124 |
| This theorem is referenced by: rexanuz2nf 45460 |
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