|   | 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 5152 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | 
| 4 | 1, 3 | mtbird 325 | 1 ⊢ (𝜑 → ¬ 𝐴𝑅𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 class class class wbr 5142 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 | 
| This theorem is referenced by: supgtoreq 9511 rlimno1 15691 pczndvds 16904 pcadd 16928 recld2 24837 itg2cnlem2 25798 dgrub 26274 gausslemma2dlem1a 27410 nosupbnd1lem1 27754 nosupbnd2lem1 27761 noinfbnd1lem1 27769 noinfbnd2 27777 mirbtwnhl 28689 sqrtcval 43659 | 
| Copyright terms: Public domain | W3C validator |