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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 2697 | . 2 | |
2 | elex 2697 | . 2 | |
3 | sbccom 2984 | . . . . . 6 | |
4 | 3 | a1i 9 | . . . . 5 |
5 | sbcel2g 3023 | . . . . . . 7 | |
6 | 5 | sbcbidv 2967 | . . . . . 6 |
7 | 6 | adantl 275 | . . . . 5 |
8 | sbcel2g 3023 | . . . . . . 7 | |
9 | 8 | sbcbidv 2967 | . . . . . 6 |
10 | 9 | adantr 274 | . . . . 5 |
11 | 4, 7, 10 | 3bitr3d 217 | . . . 4 |
12 | sbcel2g 3023 | . . . . 5 | |
13 | 12 | adantr 274 | . . . 4 |
14 | sbcel2g 3023 | . . . . 5 | |
15 | 14 | adantl 275 | . . . 4 |
16 | 11, 13, 15 | 3bitr3d 217 | . . 3 |
17 | 16 | eqrdv 2137 | . 2 |
18 | 1, 2, 17 | syl2an 287 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cvv 2686 wsbc 2909 csb 3003 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sbc 2910 df-csb 3004 |
This theorem is referenced by: ovmpos 5894 |
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