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Theorem csbuni 4250
 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 3831 . . . 4
2 sbcex2 3356 . . . . . 6
3 sbcan 3348 . . . . . . . 8
4 sbcg 3371 . . . . . . . . . 10
54anbi1d 709 . . . . . . . . 9
6 sbcel2 3812 . . . . . . . . . 10
76anbi2i 698 . . . . . . . . 9
85, 7syl6bb 264 . . . . . . . 8
93, 8syl5bb 260 . . . . . . 7
109exbidv 1761 . . . . . 6
112, 10syl5bb 260 . . . . 5
1211abbidv 2565 . . . 4
131, 12syl5eq 2482 . . 3
14 df-uni 4223 . . . 4
1514csbeq2i 3816 . . 3
16 df-uni 4223 . . 3
1713, 15, 163eqtr4g 2495 . 2
18 csbprc 3804 . . 3
19 csbprc 3804 . . . . 5
2019unieqd 4232 . . . 4
21 uni0 4249 . . . 4
2220, 21syl6req 2487 . . 3
2318, 22eqtrd 2470 . 2
2417, 23pm2.61i 167 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wa 370   wceq 1437  wex 1659   wcel 1870  cab 2414  cvv 3087  wsbc 3305  csb 3401  c0 3767  cuni 4222 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-in 3449  df-ss 3456  df-nul 3768  df-sn 4003  df-uni 4223 This theorem is referenced by: (None)
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