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Mirrors > Home > ILE Home > Th. List > ceqex | Unicode version |
Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) |
Ref | Expression |
---|---|
ceqex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 1482 | . . 3 | |
2 | isset 2561 | . . 3 | |
3 | 1, 2 | sylibr 137 | . 2 |
4 | eqeq2 2049 | . . . 4 | |
5 | 4 | anbi1d 438 | . . . . . 6 |
6 | 5 | exbidv 1706 | . . . . 5 |
7 | 6 | bibi2d 221 | . . . 4 |
8 | 4, 7 | imbi12d 223 | . . 3 |
9 | 19.8a 1482 | . . . . 5 | |
10 | 9 | ex 108 | . . . 4 |
11 | vex 2560 | . . . . . 6 | |
12 | 11 | alexeq 2670 | . . . . 5 |
13 | sp 1401 | . . . . . 6 | |
14 | 13 | com12 27 | . . . . 5 |
15 | 12, 14 | syl5bir 142 | . . . 4 |
16 | 10, 15 | impbid 120 | . . 3 |
17 | 8, 16 | vtoclg 2613 | . 2 |
18 | 3, 17 | mpcom 32 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wex 1381 wcel 1393 cvv 2557 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: ceqsexg 2672 sbc6g 2788 |
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