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Mirrors > Home > ILE Home > Th. List > csbxpg | Unicode version |
Description: Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) |
Ref | Expression |
---|---|
csbxpg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbabg 3110 | . . 3 | |
2 | sbcexg 3009 | . . . . 5 | |
3 | sbcexg 3009 | . . . . . . 7 | |
4 | sbcang 2998 | . . . . . . . . 9 | |
5 | sbcg 3024 | . . . . . . . . . 10 | |
6 | sbcang 2998 | . . . . . . . . . . 11 | |
7 | sbcel2g 3070 | . . . . . . . . . . . 12 | |
8 | sbcel2g 3070 | . . . . . . . . . . . 12 | |
9 | 7, 8 | anbi12d 470 | . . . . . . . . . . 11 |
10 | 6, 9 | bitrd 187 | . . . . . . . . . 10 |
11 | 5, 10 | anbi12d 470 | . . . . . . . . 9 |
12 | 4, 11 | bitrd 187 | . . . . . . . 8 |
13 | 12 | exbidv 1818 | . . . . . . 7 |
14 | 3, 13 | bitrd 187 | . . . . . 6 |
15 | 14 | exbidv 1818 | . . . . 5 |
16 | 2, 15 | bitrd 187 | . . . 4 |
17 | 16 | abbidv 2288 | . . 3 |
18 | 1, 17 | eqtrd 2203 | . 2 |
19 | df-xp 4617 | . . . 4 | |
20 | df-opab 4051 | . . . 4 | |
21 | 19, 20 | eqtri 2191 | . . 3 |
22 | 21 | csbeq2i 3076 | . 2 |
23 | df-xp 4617 | . . 3 | |
24 | df-opab 4051 | . . 3 | |
25 | 23, 24 | eqtri 2191 | . 2 |
26 | 18, 22, 25 | 3eqtr4g 2228 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wex 1485 wcel 2141 cab 2156 wsbc 2955 csb 3049 cop 3586 copab 4049 cxp 4609 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-sbc 2956 df-csb 3050 df-opab 4051 df-xp 4617 |
This theorem is referenced by: csbresg 4894 |
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