Theorem csbabg 2963

Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
csbabg	|- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )

Distinct variable groups:   y, A   x, y

Allowed substitution hints:   ph( x, y)   A( x)   V( x, y)

Proof of Theorem csbabg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbccom 2889	. . . 4 |- ( [. z / y ]. [. A / x ]. ph <-> [. A / x ]. [. z / y ]. ph )
2		df-clab 2068	. . . . 5 |- ( z e. { y | [. A / x ]. ph } <-> [ z / y ] [. A / x ]. ph )
3		sbsbc 2819	. . . . 5 |- ( [ z / y ] [. A / x ]. ph <-> [. z / y ]. [. A / x ]. ph )
4	2, 3	bitri 182	. . . 4 |- ( z e. { y | [. A / x ]. ph } <-> [. z / y ]. [. A / x ]. ph )
5		df-clab 2068	. . . . . 6 |- ( z e. { y | ph } <-> [ z / y ] ph )
6		sbsbc 2819	. . . . . 6 |- ( [ z / y ] ph <-> [. z / y ]. ph )
7	5, 6	bitri 182	. . . . 5 |- ( z e. { y | ph } <-> [. z / y ]. ph )
8	7	sbcbii 2873	. . . 4 |- ( [. A / x ]. z e. { y | ph } <-> [. A / x ]. [. z / y ]. ph )
9	1, 4, 8	3bitr4i 210	. . 3 |- ( z e. { y | [. A / x ]. ph } <-> [. A / x ]. z e. { y | ph } )
10		sbcel2g 2927	. . 3 |- ( A e. V -> ( [. A / x ]. z e. { y | ph } <-> z e. [_ A / x ]_ { y | ph } ) )
11	9, 10	syl5rbb 191	. 2 |- ( A e. V -> ( z e. [_ A / x ]_ { y | ph } <-> z e. { y | [. A / x ]. ph } ) )
12	11	eqrdv 2079	1 |- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   [wsb 1685   {cab 2067   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  csbsng  3453  csbunig  3609  csbxpg  4439  csbdmg  4547  csbrng  4802