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Theorem csbxp 5090
 Description: Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbxp

Proof of Theorem csbxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 3860 . . 3
2 sbcex2 3381 . . . . 5
3 sbcex2 3381 . . . . . . 7
4 sbcan 3370 . . . . . . . . 9
5 sbcg 3399 . . . . . . . . . . 11
6 sbcan 3370 . . . . . . . . . . . . 13
7 sbcel2 3839 . . . . . . . . . . . . . 14
8 sbcel2 3839 . . . . . . . . . . . . . 14
97, 8anbi12i 697 . . . . . . . . . . . . 13
106, 9bitri 249 . . . . . . . . . . . 12
1110a1i 11 . . . . . . . . . . 11
125, 11anbi12d 710 . . . . . . . . . 10
13 sbcex 3337 . . . . . . . . . . . . 13
1413con3i 135 . . . . . . . . . . . 12
1514intnand 916 . . . . . . . . . . 11
16 noel 3797 . . . . . . . . . . . . . . 15
1716a1i 11 . . . . . . . . . . . . . 14
18 csbprc 3830 . . . . . . . . . . . . . 14
1917, 18neleqtrrd 2570 . . . . . . . . . . . . 13
2019intnand 916 . . . . . . . . . . . 12
2120intnand 916 . . . . . . . . . . 11
2215, 212falsed 351 . . . . . . . . . 10
2312, 22pm2.61i 164 . . . . . . . . 9
244, 23bitri 249 . . . . . . . 8
2524exbii 1668 . . . . . . 7
263, 25bitri 249 . . . . . 6
2726exbii 1668 . . . . 5
282, 27bitri 249 . . . 4
2928abbii 2591 . . 3
301, 29eqtri 2486 . 2
31 df-xp 5014 . . . 4
32 df-opab 4516 . . . 4
3331, 32eqtri 2486 . . 3
3433csbeq2i 3844 . 2
35 df-xp 5014 . . 3
36 df-opab 4516 . . 3
3735, 36eqtri 2486 . 2
3830, 34, 373eqtr4i 2496 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 184   wa 369   wceq 1395  wex 1613   wcel 1819  cab 2442  cvv 3109  wsbc 3327  csb 3430  c0 3793  cop 4038  copab 4514   cxp 5006 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794  df-opab 4516  df-xp 5014 This theorem is referenced by:  csbres  5286
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