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Theorem csbuni 4277
 Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 22-Aug-2018.)
Assertion
Ref Expression
csbuni

Proof of Theorem csbuni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 3855 . . . 4
2 sbcex2 3381 . . . . . 6
3 sbcan 3370 . . . . . . . 8
4 sbcg 3401 . . . . . . . . . 10
54anbi1d 704 . . . . . . . . 9
6 sbcel2 3831 . . . . . . . . . 10
76anbi2i 694 . . . . . . . . 9
85, 7syl6bb 261 . . . . . . . 8
93, 8syl5bb 257 . . . . . . 7
109exbidv 1714 . . . . . 6
112, 10syl5bb 257 . . . . 5
1211abbidv 2593 . . . 4
131, 12syl5eq 2510 . . 3
14 df-uni 4250 . . . 4
1514csbeq2i 3836 . . 3
16 df-uni 4250 . . 3
1713, 15, 163eqtr4g 2523 . 2
18 csbprc 3821 . . 3
19 csbprc 3821 . . . . 5
2019unieqd 4259 . . . 4
21 uni0 4276 . . . 4
2220, 21syl6req 2515 . . 3
2318, 22eqtrd 2498 . 2
2417, 23pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784  U.cuni 4249 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-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-uni 4250
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