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Theorem csbxp 5086
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 3855 . . 3
2 sbcex2 3381 . . . . 5
3 sbcex2 3381 . . . . . . 7
4 sbcan 3370 . . . . . . . . 9
5 sbcg 3401 . . . . . . . . . . 11
6 sbcan 3370 . . . . . . . . . . . . 13
7 sbcel2 3831 . . . . . . . . . . . . . 14
8 sbcel2 3831 . . . . . . . . . . . . . 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 3788 . . . . . . . . . . . . . . 15
1716a1i 11 . . . . . . . . . . . . . 14
18 csbprc 3821 . . . . . . . . . . . . . 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 1667 . . . . . . 7
263, 25bitri 249 . . . . . 6
2726exbii 1667 . . . . 5
282, 27bitri 249 . . . 4
2928abbii 2591 . . 3
301, 29eqtri 2486 . 2
31 df-xp 5010 . . . 4
32 df-opab 4511 . . . 4
3331, 32eqtri 2486 . . 3
3433csbeq2i 3836 . 2
35 df-xp 5010 . . 3
36 df-opab 4511 . . 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  E.wex 1612  e.wcel 1818  {cab 2442   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784  <.cop 4035  {copab 4509  X.cxp 5002
This theorem is referenced by:  csbres  5281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  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 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-opab 4511  df-xp 5010
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