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Mirrors > Home > MPE Home > Th. List > ertr2d | 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) |
ertrd.5 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
ertrd.6 | ⊢ (𝜑 → 𝐵𝑅𝐶) |
Ref | Expression |
---|---|
ertr2d | ⊢ (𝜑 → 𝐶𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersymb.1 | . 2 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
2 | ertrd.5 | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
3 | ertrd.6 | . . 3 ⊢ (𝜑 → 𝐵𝑅𝐶) | |
4 | 1, 2, 3 | ertrd 8155 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐶) |
5 | 1, 4 | ersym 8151 | 1 ⊢ (𝜑 → 𝐶𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 4962 Er wer 8136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-er 8139 |
This theorem is referenced by: pi1xfrcnvlem 23343 |
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