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Mirrors > Home > NFE 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 3197 | . . 3 | |
2 | sbcexg 3096 | . . . . 5 | |
3 | sbcexg 3096 | . . . . . . 7 | |
4 | sbcang 3089 | . . . . . . . . 9 | |
5 | sbcg 3111 | . . . . . . . . . 10 | |
6 | sbcang 3089 | . . . . . . . . . . 11 | |
7 | sbcel2g 3157 | . . . . . . . . . . . 12 | |
8 | sbcel2g 3157 | . . . . . . . . . . . 12 | |
9 | 7, 8 | anbi12d 691 | . . . . . . . . . . 11 |
10 | 6, 9 | bitrd 244 | . . . . . . . . . 10 |
11 | 5, 10 | anbi12d 691 | . . . . . . . . 9 |
12 | 4, 11 | bitrd 244 | . . . . . . . 8 |
13 | 12 | exbidv 1626 | . . . . . . 7 |
14 | 3, 13 | bitrd 244 | . . . . . 6 |
15 | 14 | exbidv 1626 | . . . . 5 |
16 | 2, 15 | bitrd 244 | . . . 4 |
17 | 16 | abbidv 2467 | . . 3 |
18 | 1, 17 | eqtrd 2385 | . 2 |
19 | df-xp 4784 | . . . 4 | |
20 | df-opab 4623 | . . . 4 | |
21 | 19, 20 | eqtri 2373 | . . 3 |
22 | 21 | csbeq2i 3162 | . 2 |
23 | df-xp 4784 | . . 3 | |
24 | df-opab 4623 | . . 3 | |
25 | 23, 24 | eqtri 2373 | . 2 |
26 | 18, 22, 25 | 3eqtr4g 2410 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wex 1541 wceq 1642 wcel 1710 cab 2339 wsbc 3046 csb 3136 cop 4561 copab 4622 cxp 4770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-sbc 3047 df-csb 3137 df-opab 4623 df-xp 4784 |
This theorem is referenced by: csbresg 4976 |
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