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Theorem csbxpg 4660
 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 3088 . . 3
2 sbcexg 2987 . . . . 5
3 sbcexg 2987 . . . . . . 7
4 sbcang 2976 . . . . . . . . 9
5 sbcg 3002 . . . . . . . . . 10
6 sbcang 2976 . . . . . . . . . . 11
7 sbcel2g 3048 . . . . . . . . . . . 12
8 sbcel2g 3048 . . . . . . . . . . . 12
97, 8anbi12d 465 . . . . . . . . . . 11
106, 9bitrd 187 . . . . . . . . . 10
115, 10anbi12d 465 . . . . . . . . 9
124, 11bitrd 187 . . . . . . . 8
1312exbidv 1802 . . . . . . 7
143, 13bitrd 187 . . . . . 6
1514exbidv 1802 . . . . 5
162, 15bitrd 187 . . . 4
1716abbidv 2272 . . 3
181, 17eqtrd 2187 . 2
19 df-xp 4585 . . . 4
20 df-opab 4022 . . . 4
2119, 20eqtri 2175 . . 3
2221csbeq2i 3054 . 2
23 df-xp 4585 . . 3
24 df-opab 4022 . . 3
2523, 24eqtri 2175 . 2
2618, 22, 253eqtr4g 2212 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1332  wex 1469   wcel 2125  cab 2140  wsbc 2933  csb 3027  cop 3559  copab 4020   cxp 4577 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-sbc 2934  df-csb 3028  df-opab 4022  df-xp 4585 This theorem is referenced by:  csbresg  4862
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