Theorem csbabg 3056

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 2979	. . . 4 |- ( [. z / y ]. [. A / x ]. ph <-> [. A / x ]. [. z / y ]. ph )
2		df-clab 2124	. . . . 5 |- ( z e. { y | [. A / x ]. ph } <-> [ z / y ] [. A / x ]. ph )
3		sbsbc 2908	. . . . 5 |- ( [ z / y ] [. A / x ]. ph <-> [. z / y ]. [. A / x ]. ph )
4	2, 3	bitri 183	. . . 4 |- ( z e. { y | [. A / x ]. ph } <-> [. z / y ]. [. A / x ]. ph )
5		df-clab 2124	. . . . . 6 |- ( z e. { y | ph } <-> [ z / y ] ph )
6		sbsbc 2908	. . . . . 6 |- ( [ z / y ] ph <-> [. z / y ]. ph )
7	5, 6	bitri 183	. . . . 5 |- ( z e. { y | ph } <-> [. z / y ]. ph )
8	7	sbcbii 2963	. . . 4 |- ( [. A / x ]. z e. { y | ph } <-> [. A / x ]. [. z / y ]. ph )
9	1, 4, 8	3bitr4i 211	. . 3 |- ( z e. { y | [. A / x ]. ph } <-> [. A / x ]. z e. { y | ph } )
10		sbcel2g 3018	. . 3 |- ( A e. V -> ( [. A / x ]. z e. { y | ph } <-> z e. [_ A / x ]_ { y | ph } ) )
11	9, 10	syl5rbb 192	. 2 |- ( A e. V -> ( z e. [_ A / x ]_ { y | ph } <-> z e. { y | [. A / x ]. ph } ) )
12	11	eqrdv 2135	1 |- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   [wsb 1735   {cab 2123   [.wsbc 2904   [_csb 2998

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by:  csbsng  3579  csbunig  3739  csbxpg  4615  csbdmg  4728  csbrng  4995