Theorem csbsng 3696

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 3157	. . 3 |- ( A e. V -> [_ A / x ]_ { y | y = B } = { y | [. A / x ]. y = B } )
2		sbceq2g 3117	. . . 4 |- ( A e. V -> ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) )
3	2	abbidv 2324	. . 3 |- ( A e. V -> { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } )
4	1, 3	eqtrd 2239	. 2 |- ( A e. V -> [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } )
5		df-sn 3641	. . 3 |- { B } = { y | y = B }
6	5	csbeq2i 3122	. 2 |- [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B }
7		df-sn 3641	. 2 |- { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B }
8	4, 6, 7	3eqtr4g 2264	1 |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   {cab 2192   [.wsbc 3000   [_csb 3095   {csn 3635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3001  df-csb 3096  df-sn 3641

This theorem is referenced by: (None)