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Theorem csbxpg 4905
 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 3310 . . 3
2 sbcexg 3211 . . . . 5
3 sbcexg 3211 . . . . . . 7
4 sbcang 3204 . . . . . . . . 9
5 sbcg 3226 . . . . . . . . . 10
6 sbcang 3204 . . . . . . . . . . 11
7 sbcel2g 3272 . . . . . . . . . . . 12
8 sbcel2g 3272 . . . . . . . . . . . 12
97, 8anbi12d 692 . . . . . . . . . . 11
106, 9bitrd 245 . . . . . . . . . 10
115, 10anbi12d 692 . . . . . . . . 9
124, 11bitrd 245 . . . . . . . 8
1312exbidv 1636 . . . . . . 7
143, 13bitrd 245 . . . . . 6
1514exbidv 1636 . . . . 5
162, 15bitrd 245 . . . 4
1716abbidv 2550 . . 3
181, 17eqtrd 2468 . 2
19 df-xp 4884 . . . 4
20 df-opab 4267 . . . 4
2119, 20eqtri 2456 . . 3
2221csbeq2i 3277 . 2
23 df-xp 4884 . . 3
24 df-opab 4267 . . 3
2523, 24eqtri 2456 . 2
2618, 22, 253eqtr4g 2493 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1550   wceq 1652   wcel 1725  cab 2422  wsbc 3161  csb 3251  cop 3817  copab 4265   cxp 4876 This theorem is referenced by:  csbresg  5149  csbresgVD  29007 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252  df-opab 4267  df-xp 4884
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