Theorem csbabg 3142

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 3061	. . . 4 |- ( [. z / y ]. [. A / x ]. ph <-> [. A / x ]. [. z / y ]. ph )
2		df-clab 2180	. . . . 5 |- ( z e. { y | [. A / x ]. ph } <-> [ z / y ] [. A / x ]. ph )
3		sbsbc 2989	. . . . 5 |- ( [ z / y ] [. A / x ]. ph <-> [. z / y ]. [. A / x ]. ph )
4	2, 3	bitri 184	. . . 4 |- ( z e. { y | [. A / x ]. ph } <-> [. z / y ]. [. A / x ]. ph )
5		df-clab 2180	. . . . . 6 |- ( z e. { y | ph } <-> [ z / y ] ph )
6		sbsbc 2989	. . . . . 6 |- ( [ z / y ] ph <-> [. z / y ]. ph )
7	5, 6	bitri 184	. . . . 5 |- ( z e. { y | ph } <-> [. z / y ]. ph )
8	7	sbcbii 3045	. . . 4 |- ( [. A / x ]. z e. { y | ph } <-> [. A / x ]. [. z / y ]. ph )
9	1, 4, 8	3bitr4i 212	. . 3 |- ( z e. { y | [. A / x ]. ph } <-> [. A / x ]. z e. { y | ph } )
10		sbcel2g 3101	. . 3 |- ( A e. V -> ( [. A / x ]. z e. { y | ph } <-> z e. [_ A / x ]_ { y | ph } ) )
11	9, 10	bitr2id 193	. 2 |- ( A e. V -> ( z e. [_ A / x ]_ { y | ph } <-> z e. { y | [. A / x ]. ph } ) )
12	11	eqrdv 2191	1 |- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )

Syntax hints:   -> wi 4   = wceq 1364   [wsb 1773   e. wcel 2164   {cab 2179   [.wsbc 2985   [_csb 3080

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-csb 3081

This theorem is referenced by:  csbsng  3679  csbunig  3843  csbxpg  4740  csbdmg  4856  csbrng  5127  csbwrdg  10943