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Mirrors > Home > ILE Home > Th. List > ertr2d | 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 6544 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐶) |
5 | 1, 4 | ersym 6540 | 1 ⊢ (𝜑 → 𝐶𝑅𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 class class class wbr 4000 Er wer 6525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-er 6528 |
This theorem is referenced by: (None) |
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