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Mirrors > Home > ILE Home > Th. List > ecoptocl | Unicode version |
Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
ecoptocl.1 | |
ecoptocl.2 | |
ecoptocl.3 |
Ref | Expression |
---|---|
ecoptocl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 6481 | . . 3 | |
2 | eqid 2139 | . . . . 5 | |
3 | eceq1 6464 | . . . . . . 7 | |
4 | 3 | eqeq2d 2151 | . . . . . 6 |
5 | 4 | imbi1d 230 | . . . . 5 |
6 | ecoptocl.3 | . . . . . 6 | |
7 | ecoptocl.2 | . . . . . . 7 | |
8 | 7 | eqcoms 2142 | . . . . . 6 |
9 | 6, 8 | syl5ibcom 154 | . . . . 5 |
10 | 2, 5, 9 | optocl 4615 | . . . 4 |
11 | 10 | rexlimiv 2543 | . . 3 |
12 | 1, 11 | syl 14 | . 2 |
13 | ecoptocl.1 | . 2 | |
14 | 12, 13 | eleq2s 2234 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2417 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: 2ecoptocl 6517 3ecoptocl 6518 mulidnq 7197 recexnq 7198 ltsonq 7206 distrnq0 7267 addassnq0 7270 ltposr 7571 0idsr 7575 1idsr 7576 00sr 7577 recexgt0sr 7581 archsr 7590 srpospr 7591 map2psrprg 7613 |
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