| 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 5155 | . 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 1540 class class class wbr 5143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 |
| This theorem is referenced by: rexanuz2nf 45503 |
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