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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 8161 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
5 | 1, 2, 4 | mp2and 695 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 4968 Er wer 8143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-br 4969 df-opab 5031 df-xp 5456 df-rel 5457 df-co 5459 df-er 8146 |
This theorem is referenced by: ertr2d 8163 ertr3d 8164 ertr4d 8165 erinxp 8228 nqereq 10210 adderpq 10231 mulerpq 10232 efgred2 18610 efgcpbllemb 18612 efgcpbl2 18614 pcophtb 23320 pi1xfr 23346 pi1xfrcnvlem 23347 erbr3b 30054 |
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