Theorem csbsng 4030

Description: Distribute proper substitution through the singleton of a class. csbsng 4030 is derived from the virtual deduction proof csbsngVD 37290. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
csbsng	|- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )

Proof of Theorem csbsng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbab 3797	. . 3 |- [_ A / x ]_ { y | y = B } = { y | [. A / x ]. y = B }
2		sbceq2g 3779	. . . 4 |- ( A e. V -> ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) )
3	2	abbidv 2569	. . 3 |- ( A e. V -> { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } )
4	1, 3	syl5eq 2497	. 2 |- ( A e. V -> [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } )
5		df-sn 3969	. . 3 |- { B } = { y | y = B }
6	5	csbeq2i 3782	. 2 |- [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B }
7		df-sn 3969	. 2 |- { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B }
8	4, 6, 7	3eqtr4g 2510	1 |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )

Syntax hints:   -> wi 4   = wceq 1444   e. wcel 1887   {cab 2437   [.wsbc 3267   [_csb 3363   {csn 3968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969

This theorem is referenced by:  csbprg  4031  csbopg2  31725  csbpredg  31727  csbfv12gALTOLD  37213  csbfv12gALTVD  37296