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Theorem csbxpg 4364
 Description: Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbxpg

Proof of Theorem csbxpg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 2901 . . 3
2 sbcexg 2807 . . . . 5
3 sbcexg 2807 . . . . . . 7
4 sbcang 2800 . . . . . . . . 9
5 sbcg 2821 . . . . . . . . . 10
6 sbcang 2800 . . . . . . . . . . 11
7 sbcel2g 2865 . . . . . . . . . . . 12
8 sbcel2g 2865 . . . . . . . . . . . 12
97, 8anbi12d 442 . . . . . . . . . . 11
106, 9bitrd 177 . . . . . . . . . 10
115, 10anbi12d 442 . . . . . . . . 9
124, 11bitrd 177 . . . . . . . 8
1312exbidv 1703 . . . . . . 7
143, 13bitrd 177 . . . . . 6
1514exbidv 1703 . . . . 5
162, 15bitrd 177 . . . 4
1716abbidv 2152 . . 3
181, 17eqtrd 2069 . 2
19 df-xp 4294 . . . 4
20 df-opab 3810 . . . 4
2119, 20eqtri 2057 . . 3
2221csbeq2i 2870 . 2
23 df-xp 4294 . . 3
24 df-opab 3810 . . 3
2523, 24eqtri 2057 . 2
2618, 22, 253eqtr4g 2094 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390  cab 2023  wsbc 2758  csb 2846  cop 3370  copab 3808   cxp 4286 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-opab 3810  df-xp 4294 This theorem is referenced by:  csbresg  4558
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