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Mirrors > Home > ILE Home > Th. List > 3ecoptocl | Unicode version |
Description: Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.) |
Ref | Expression |
---|---|
3ecoptocl.1 | |
3ecoptocl.2 | |
3ecoptocl.3 | |
3ecoptocl.4 | |
3ecoptocl.5 |
Ref | Expression |
---|---|
3ecoptocl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3ecoptocl.1 | . . . 4 | |
2 | 3ecoptocl.3 | . . . . 5 | |
3 | 2 | imbi2d 229 | . . . 4 |
4 | 3ecoptocl.4 | . . . . 5 | |
5 | 4 | imbi2d 229 | . . . 4 |
6 | 3ecoptocl.2 | . . . . . . 7 | |
7 | 6 | imbi2d 229 | . . . . . 6 |
8 | 3ecoptocl.5 | . . . . . . 7 | |
9 | 8 | 3expib 1184 | . . . . . 6 |
10 | 1, 7, 9 | ecoptocl 6516 | . . . . 5 |
11 | 10 | com12 30 | . . . 4 |
12 | 1, 3, 5, 11 | 2ecoptocl 6517 | . . 3 |
13 | 12 | com12 30 | . 2 |
14 | 13 | 3impib 1179 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cop 3530 cxp 4537 cec 6427 cqs 6428 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-ec 6431 df-qs 6435 |
This theorem is referenced by: ecovass 6538 ecoviass 6539 ecovdi 6540 ecovidi 6541 ltsonq 7206 ltanqg 7208 ltmnqg 7209 lttrsr 7570 ltsosr 7572 ltasrg 7578 mulextsr1 7589 |
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