![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4940 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
4 | 1, 3 | mtbird 317 | 1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1507 class class class wbr 4930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-br 4931 |
This theorem is referenced by: supgtoreq 8731 rlimno1 14874 pczndvds 16060 pcadd 16084 recld2 23128 itg2cnlem2 24069 dgrub 24530 gausslemma2dlem1a 25646 mirbtwnhl 26171 noprefixmo 32723 nosupbnd1lem1 32729 nosupbnd2lem1 32736 |
Copyright terms: Public domain | W3C validator |