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Theorem csbxpg 4813
 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 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
97, 8anbi12d 691 . . . . . . . . . . 11
106, 9bitrd 244 . . . . . . . . . 10
115, 10anbi12d 691 . . . . . . . . 9
124, 11bitrd 244 . . . . . . . 8
1312exbidv 1626 . . . . . . 7
143, 13bitrd 244 . . . . . 6
1514exbidv 1626 . . . . 5
162, 15bitrd 244 . . . 4
1716abbidv 2467 . . 3
181, 17eqtrd 2385 . 2
19 df-xp 4784 . . . 4
20 df-opab 4623 . . . 4
2119, 20eqtri 2373 . . 3
2221csbeq2i 3162 . 2
23 df-xp 4784 . . 3
24 df-opab 4623 . . 3
2523, 24eqtri 2373 . 2
2618, 22, 253eqtr4g 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-1 5  ax-2 6  ax-3 7  ax-mp 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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