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Mirrors > Home > ILE Home > Th. List > ecovass | Unicode version |
Description: Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6539 will be more useful. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
Ref | Expression |
---|---|
ecovass.1 | |
ecovass.2 | |
ecovass.3 | |
ecovass.4 | |
ecovass.5 | |
ecovass.6 | |
ecovass.7 | |
ecovass.8 | |
ecovass.9 |
Ref | Expression |
---|---|
ecovass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovass.1 | . 2 | |
2 | oveq1 5781 | . . . 4 | |
3 | 2 | oveq1d 5789 | . . 3 |
4 | oveq1 5781 | . . 3 | |
5 | 3, 4 | eqeq12d 2154 | . 2 |
6 | oveq2 5782 | . . . 4 | |
7 | 6 | oveq1d 5789 | . . 3 |
8 | oveq1 5781 | . . . 4 | |
9 | 8 | oveq2d 5790 | . . 3 |
10 | 7, 9 | eqeq12d 2154 | . 2 |
11 | oveq2 5782 | . . 3 | |
12 | oveq2 5782 | . . . 4 | |
13 | 12 | oveq2d 5790 | . . 3 |
14 | 11, 13 | eqeq12d 2154 | . 2 |
15 | ecovass.8 | . . . 4 | |
16 | ecovass.9 | . . . 4 | |
17 | opeq12 3707 | . . . . 5 | |
18 | 17 | eceq1d 6465 | . . . 4 |
19 | 15, 16, 18 | mp2an 422 | . . 3 |
20 | ecovass.2 | . . . . . . 7 | |
21 | 20 | oveq1d 5789 | . . . . . 6 |
22 | 21 | adantr 274 | . . . . 5 |
23 | ecovass.6 | . . . . . 6 | |
24 | ecovass.4 | . . . . . 6 | |
25 | 23, 24 | sylan 281 | . . . . 5 |
26 | 22, 25 | eqtrd 2172 | . . . 4 |
27 | 26 | 3impa 1176 | . . 3 |
28 | ecovass.3 | . . . . . . 7 | |
29 | 28 | oveq2d 5790 | . . . . . 6 |
30 | 29 | adantl 275 | . . . . 5 |
31 | ecovass.7 | . . . . . 6 | |
32 | ecovass.5 | . . . . . 6 | |
33 | 31, 32 | sylan2 284 | . . . . 5 |
34 | 30, 33 | eqtrd 2172 | . . . 4 |
35 | 34 | 3impb 1177 | . . 3 |
36 | 19, 27, 35 | 3eqtr4a 2198 | . 2 |
37 | 1, 5, 10, 14, 36 | 3ecoptocl 6518 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 cop 3530 cxp 4537 (class class class)co 5774 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-uni 3737 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-iota 5088 df-fv 5131 df-ov 5777 df-ec 6431 df-qs 6435 |
This theorem is referenced by: (None) |
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