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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 8300 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) |
4 | ertr3d.6 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) | |
5 | 1, 3, 4 | ertrd 8304 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5065 Er wer 8285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-er 8288 |
This theorem is referenced by: nqereq 10356 efgred2 18878 xmetresbl 23046 pcophtb 23632 pi1xfr 23658 pi1xfrcnvlem 23659 erbr3b 30367 prtlem10 36000 |
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