Theorem csbsng 4021

Description: Distribute proper substitution through the singleton of a class. csbsng 4021 is derived from the virtual deduction proof csbsngVD 37353. (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 3801	. . 3 |- [_ A / x ]_ { y | y = B } = { y | [. A / x ]. y = B }
2		sbceq2g 3783	. . . 4 |- ( A e. V -> ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) )
3	2	abbidv 2589	. . 3 |- ( A e. V -> { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } )
4	1, 3	syl5eq 2517	. 2 |- ( A e. V -> [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } )
5		df-sn 3960	. . 3 |- { B } = { y | y = B }
6	5	csbeq2i 3786	. 2 |- [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B }
7		df-sn 3960	. 2 |- { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B }
8	4, 6, 7	3eqtr4g 2530	1 |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )

Syntax hints:   -> wi 4   = wceq 1452   e. wcel 1904   {cab 2457   [.wsbc 3255   [_csb 3349   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-sn 3960

This theorem is referenced by:  csbprg  4022  csbopg2  31795  csbpredg  31797  csbfv12gALTOLD  37276  csbfv12gALTVD  37359