Theorem csbsng 4243

Description: Distribute proper substitution through the singleton of a class. csbsng 4243 is derived from the virtual deduction proof csbsngVD 39129. (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 4008	. . 3 |- [_ A / x ]_ { y | y = B } = { y | [. A / x ]. y = B }
2		sbceq2g 3990	. . . 4 |- ( A e. V -> ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) )
3	2	abbidv 2741	. . 3 |- ( A e. V -> { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } )
4	1, 3	syl5eq 2668	. 2 |- ( A e. V -> [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } )
5		df-sn 4178	. . 3 |- { B } = { y | y = B }
6	5	csbeq2i 3993	. 2 |- [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B }
7		df-sn 4178	. 2 |- { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B }
8	4, 6, 7	3eqtr4g 2681	1 |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916  df-sn 4178

This theorem is referenced by:  csbprg  4244  csbopg  4420  csbpredg  33172  csbfv12gALTOLD  39052  csbfv12gALTVD  39135