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Mirrors > Home > ILE Home > Th. List > sbequi | Unicode version |
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.) |
Ref | Expression |
---|---|
sbequi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsb2or 1758 | . . . 4 | |
2 | nfr 1451 | . . . . . 6 | |
3 | equvini 1681 | . . . . . . 7 | |
4 | stdpc7 1693 | . . . . . . . . 9 | |
5 | sbequ1 1691 | . . . . . . . . 9 | |
6 | 4, 5 | sylan9 401 | . . . . . . . 8 |
7 | 6 | eximi 1531 | . . . . . . 7 |
8 | 19.35-1 1555 | . . . . . . 7 | |
9 | 3, 7, 8 | 3syl 17 | . . . . . 6 |
10 | 2, 9 | syl9 71 | . . . . 5 |
11 | 10 | orim2i 710 | . . . 4 |
12 | 1, 11 | ax-mp 7 | . . 3 |
13 | nfsb2or 1758 | . . . . 5 | |
14 | 19.9t 1573 | . . . . . . 7 | |
15 | 14 | biimpd 142 | . . . . . 6 |
16 | 15 | orim2i 710 | . . . . 5 |
17 | 13, 16 | ax-mp 7 | . . . 4 |
18 | ax-1 5 | . . . . 5 | |
19 | 18 | orim2i 710 | . . . 4 |
20 | 17, 19 | ax-mp 7 | . . 3 |
21 | 12, 20 | sbequilem 1759 | . 2 |
22 | sbequ2 1692 | . . . . . . 7 | |
23 | 22 | sps 1470 | . . . . . 6 |
24 | 23 | adantr 270 | . . . . 5 |
25 | sbequ1 1691 | . . . . . 6 | |
26 | drsb1 1720 | . . . . . . . 8 | |
27 | 26 | biimpd 142 | . . . . . . 7 |
28 | 27 | alequcoms 1449 | . . . . . 6 |
29 | 25, 28 | sylan9r 402 | . . . . 5 |
30 | 24, 29 | syld 44 | . . . 4 |
31 | 30 | ex 113 | . . 3 |
32 | drsb1 1720 | . . . . . . . . 9 | |
33 | 32 | biimpd 142 | . . . . . . . 8 |
34 | stdpc7 1693 | . . . . . . . 8 | |
35 | 33, 34 | sylan9 401 | . . . . . . 7 |
36 | 5 | sps 1470 | . . . . . . . 8 |
37 | 36 | adantr 270 | . . . . . . 7 |
38 | 35, 37 | syld 44 | . . . . . 6 |
39 | 38 | ex 113 | . . . . 5 |
40 | 39 | orim1i 709 | . . . 4 |
41 | pm1.2 705 | . . . 4 | |
42 | 40, 41 | syl 14 | . . 3 |
43 | 31, 42 | jaoi 668 | . 2 |
44 | 21, 43 | ax-mp 7 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wo 661 wal 1282 wnf 1389 wex 1421 wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: sbequ 1761 |
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