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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 5118 | . 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 1542 class class class wbr 5106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 |
This theorem is referenced by: rexanuz2nf 43814 |
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