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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 7754 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) |
5 | 1, 2, 4 | ertrd 7758 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 4653 Er wer 7739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-er 7742 |
This theorem is referenced by: erref 7762 erdisj 7794 nqereu 9751 nqereq 9757 efgredeu 18165 pi1xfr 22855 pi1xfrcnvlem 22856 |
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