Theorem csbsng 3644

Description: Distribute proper substitution through the singleton of a class. (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		csbabg 3110	. . 3 |- ( A e. V -> [_ A / x ]_ { y | y = B } = { y | [. A / x ]. y = B } )
2		sbceq2g 3071	. . . 4 |- ( A e. V -> ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) )
3	2	abbidv 2288	. . 3 |- ( A e. V -> { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } )
4	1, 3	eqtrd 2203	. 2 |- ( A e. V -> [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } )
5		df-sn 3589	. . 3 |- { B } = { y | y = B }
6	5	csbeq2i 3076	. 2 |- [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B }
7		df-sn 3589	. 2 |- { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B }
8	4, 6, 7	3eqtr4g 2228	1 |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   {cab 2156   [.wsbc 2955   [_csb 3049   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-sn 3589

This theorem is referenced by: (None)