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Mirrors > Home > MPE Home > Th. List > ertr3d | Structured version Visualization version GIF version |
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ersymb.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
ertr3d.5 | ⊢ (𝜑 → 𝐵𝑅𝐴) |
ertr3d.6 | ⊢ (𝜑 → 𝐵𝑅𝐶) |
Ref | Expression |
---|---|
ertr3d | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersymb.1 | . 2 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
2 | ertr3d.5 | . . 3 ⊢ (𝜑 → 𝐵𝑅𝐴) | |
3 | 1, 2 | ersym 8721 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) |
4 | ertr3d.6 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) | |
5 | 1, 3, 4 | ertrd 8725 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5148 Er wer 8706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-er 8709 |
This theorem is referenced by: nqereq 10936 efgred2 19669 xmetresbl 24263 pcophtb 24876 pi1xfr 24902 pi1xfrcnvlem 24903 erbr3b 32280 prtlem10 38201 |
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