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Mirrors > Home > MPE Home > Th. List > ertrd | 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 |
---|---|
ertrd | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ertrd.5 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | ertrd.6 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) | |
3 | ersymb.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
4 | 3 | ertr 8740 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
5 | 1, 2, 4 | mp2and 698 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5148 Er wer 8722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-co 5687 df-er 8725 |
This theorem is referenced by: ertr2d 8742 ertr3d 8743 ertr4d 8744 erinxp 8810 nqereq 10959 adderpq 10980 mulerpq 10981 efgred2 19708 efgcpbllemb 19710 efgcpbl2 19712 pcophtb 24969 pi1xfr 24995 pi1xfrcnvlem 24996 erbr3b 32420 prjspner1 42050 |
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