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Mirrors > Home > ILE Home > Th. List > eqnbrtrd | 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 3977 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
4 | 1, 3 | mtbird 663 | 1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1335 class class class wbr 3967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 |
This theorem is referenced by: pczndvds 12205 |
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