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Mirrors > Home > MPE Home > Th. List > eqnbrtrd | 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 |
---|---|
eqnbrtrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqnbrtrd.2 | ⊢ (𝜑 → ¬ 𝐵𝑅𝐶) |
Ref | Expression |
---|---|
eqnbrtrd | ⊢ (𝜑 → ¬ 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqnbrtrd.2 | . 2 ⊢ (𝜑 → ¬ 𝐵𝑅𝐶) | |
2 | eqnbrtrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | breq1d 5063 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
4 | 1, 3 | mtbird 328 | 1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 class class class wbr 5053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 |
This theorem is referenced by: supgtoreq 9086 rlimno1 15217 pczndvds 16418 pcadd 16442 recld2 23711 itg2cnlem2 24660 dgrub 25128 gausslemma2dlem1a 26246 mirbtwnhl 26771 nosupbnd1lem1 33648 nosupbnd2lem1 33655 noinfbnd1lem1 33663 noinfbnd2 33671 sqrtcval 40925 |
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