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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 3056 | . . 3 | |
2 | sbcexg 2958 | . . . . 5 | |
3 | sbcexg 2958 | . . . . . . 7 | |
4 | sbcang 2947 | . . . . . . . . 9 | |
5 | sbcg 2973 | . . . . . . . . . 10 | |
6 | sbcang 2947 | . . . . . . . . . . 11 | |
7 | sbcel2g 3018 | . . . . . . . . . . . 12 | |
8 | sbcel2g 3018 | . . . . . . . . . . . 12 | |
9 | 7, 8 | anbi12d 464 | . . . . . . . . . . 11 |
10 | 6, 9 | bitrd 187 | . . . . . . . . . 10 |
11 | 5, 10 | anbi12d 464 | . . . . . . . . 9 |
12 | 4, 11 | bitrd 187 | . . . . . . . 8 |
13 | 12 | exbidv 1797 | . . . . . . 7 |
14 | 3, 13 | bitrd 187 | . . . . . 6 |
15 | 14 | exbidv 1797 | . . . . 5 |
16 | 2, 15 | bitrd 187 | . . . 4 |
17 | 16 | abbidv 2255 | . . 3 |
18 | 1, 17 | eqtrd 2170 | . 2 |
19 | df-xp 4540 | . . . 4 | |
20 | df-opab 3985 | . . . 4 | |
21 | 19, 20 | eqtri 2158 | . . 3 |
22 | 21 | csbeq2i 3024 | . 2 |
23 | df-xp 4540 | . . 3 | |
24 | df-opab 3985 | . . 3 | |
25 | 23, 24 | eqtri 2158 | . 2 |
26 | 18, 22, 25 | 3eqtr4g 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 cab 2123 wsbc 2904 csb 2998 cop 3525 copab 3983 cxp 4532 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-sbc 2905 df-csb 2999 df-opab 3985 df-xp 4540 |
This theorem is referenced by: csbresg 4817 |
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