![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8099 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) |
4 | ertr3d.6 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) | |
5 | 1, 3, 4 | ertrd 8103 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 4925 Er wer 8084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-opab 4988 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-er 8087 |
This theorem is referenced by: nqereq 10153 efgred2 18651 xmetresbl 22765 pcophtb 23351 pi1xfr 23377 pi1xfrcnvlem 23378 erbr3b 30149 prtlem10 35483 |
Copyright terms: Public domain | W3C validator |