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Mirrors > Home > ILE Home > Th. List > ertrd | 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 6507 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
5 | 1, 2, 4 | mp2and 430 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 class class class wbr 3976 Er wer 6489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-co 4607 df-er 6492 |
This theorem is referenced by: ertr2d 6509 ertr3d 6510 ertr4d 6511 erinxp 6566 |
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