Theorem csbdm 5202

Description: Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)

Assertion

Ref	Expression
csbdm

Proof of Theorem csbdm

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbab 3855	. . 3
2		sbcex2 3381	. . . . 5
3		sbcel2 3831	. . . . . 6
4	3	exbii 1667	. . . . 5
5	2, 4	bitri 249	. . . 4
6	5	abbii 2591	. . 3
7	1, 6	eqtri 2486	. 2
8		dfdm3 5195	. . 3
9	8	csbeq2i 3836	. 2
10		dfdm3 5195	. 2
11	7, 9, 10	3eqtr4i 2496	1

Syntax hints:  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  [.wsbc 3327  [_csb 3434  <.cop 4035  domcdm 5004

This theorem is referenced by:  sbcfng  5733

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-br 4453  df-dm 5014