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Mirrors > Home > ILE Home > Th. List > breqtrd | Unicode version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
Ref | Expression |
---|---|
breqtrd.1 |
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breqtrd.2 |
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Ref | Expression |
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breqtrd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtrd.1 |
. 2
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2 | breqtrd.2 |
. . 3
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3 | 2 | breq2d 3767 |
. 2
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4 | 1, 3 | mpbid 135 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 |
This theorem is referenced by: breqtrrd 3781 syl5breq 3790 tfrexlem 5889 ltsonq 6382 addlocprlemeqgt 6515 prmuloclemcalc 6546 mullocprlem 6551 addcanprlemu 6589 ltaprlem 6591 ltaprg 6592 ltmprr 6614 cauappcvgprlemopl 6618 cauappcvgprlemloc 6624 cauappcvgprlemladdru 6628 cauappcvgprlemladdrl 6629 cauappcvgprlem1 6631 caucvgprlemm 6639 caucvgprlemopl 6640 caucvgprlemloc 6646 recexgt0sr 6701 add20 7264 mullt0 7270 ltmul1a 7375 ltm1 7593 recgt0 7597 prodgt0gt0 7598 prodgt0 7599 prodge0 7601 lemul1a 7605 recp1lt1 7646 recreclt 7647 ledivp1 7650 ltaddrp2d 8427 fz01en 8687 fzonmapblen 8813 frecfzen2 8885 ltexp2a 8960 leexp2a 8961 exple1 8964 expubnd 8965 bernneq 9022 |
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