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.

Step	Hyp	Ref	Expression
1		csbab 3855	. . . 4
2		sbcex2 3381	. . . . . 6
3		sbcan 3370	. . . . . . . 8
4		sbcg 3401	. . . . . . . . . 10
5	4	anbi1d 704	. . . . . . . . 9
6		sbcel2 3831	. . . . . . . . . 10
7	6	anbi2i 694	. . . . . . . . 9
8	5, 7	syl6bb 261	. . . . . . . 8
9	3, 8	syl5bb 257	. . . . . . 7
10	9	exbidv 1714	. . . . . 6
11	2, 10	syl5bb 257	. . . . 5
12	11	abbidv 2593	. . . 4
13	1, 12	syl5eq 2510	. . 3
14		df-uni 4250	. . . 4
15	14	csbeq2i 3836	. . 3
16		df-uni 4250	. . 3
17	13, 15, 16	3eqtr4g 2523	. 2
18		csbprc 3821	. . 3
19		csbprc 3821	. . . . 5
20	19	unieqd 4259	. . . 4
21		uni0 4276	. . . 4
22	20, 21	syl6req 2515	. . 3
23	18, 22	eqtrd 2498	. 2
24	17, 23	pm2.61i 164	1

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