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Mirrors > Home > ILE Home > Th. List > csbcomg | Unicode version |
Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) |
Ref | Expression |
---|---|
csbcomg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2610 | . 2 | |
2 | elex 2610 | . 2 | |
3 | sbccom 2889 | . . . . . 6 | |
4 | 3 | a1i 9 | . . . . 5 |
5 | sbcel2g 2927 | . . . . . . 7 | |
6 | 5 | sbcbidv 2872 | . . . . . 6 |
7 | 6 | adantl 271 | . . . . 5 |
8 | sbcel2g 2927 | . . . . . . 7 | |
9 | 8 | sbcbidv 2872 | . . . . . 6 |
10 | 9 | adantr 270 | . . . . 5 |
11 | 4, 7, 10 | 3bitr3d 216 | . . . 4 |
12 | sbcel2g 2927 | . . . . 5 | |
13 | 12 | adantr 270 | . . . 4 |
14 | sbcel2g 2927 | . . . . 5 | |
15 | 14 | adantl 271 | . . . 4 |
16 | 11, 13, 15 | 3bitr3d 216 | . . 3 |
17 | 16 | eqrdv 2079 | . 2 |
18 | 1, 2, 17 | syl2an 283 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cvv 2601 wsbc 2815 csb 2908 |
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-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 |
This theorem is referenced by: ovmpt2s 5644 |
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