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Mirrors > Home > MPE Home > Th. List > ertr4d | 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) |
ertr4d.5 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
ertr4d.6 | ⊢ (𝜑 → 𝐶𝑅𝐵) |
Ref | Expression |
---|---|
ertr4d | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersymb.1 | . 2 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
2 | ertr4d.5 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
3 | ertr4d.6 | . . 3 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
4 | 1, 3 | ersym 8659 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) |
5 | 1, 2, 4 | ertrd 8663 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5105 Er wer 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-er 8647 |
This theorem is referenced by: erref 8667 erdisj 8699 nqereu 10864 nqereq 10870 efgredeu 19532 pi1xfr 24416 pi1xfrcnvlem 24417 prjspner1 40942 |
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