Theorem csbabg 3186

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 3104	. . . 4 |- ( [. z / y ]. [. A / x ]. ph <-> [. A / x ]. [. z / y ]. ph )
2		df-clab 2216	. . . . 5 |- ( z e. { y | [. A / x ]. ph } <-> [ z / y ] [. A / x ]. ph )
3		sbsbc 3032	. . . . 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 2216	. . . . . 6 |- ( z e. { y | ph } <-> [ z / y ] ph )
6		sbsbc 3032	. . . . . 6 |- ( [ z / y ] ph <-> [. z / y ]. ph )
7	5, 6	bitri 184	. . . . 5 |- ( z e. { y | ph } <-> [. z / y ]. ph )
8	7	sbcbii 3088	. . . 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 3145	. . 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 2227	1 |- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )

Syntax hints:   -> wi 4   = wceq 1395   [wsb 1808   e. wcel 2200   {cab 2215   [.wsbc 3028   [_csb 3124

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbsng  3727  csbunig  3896  csbxpg  4800  csbdmg  4917  csbrng  5190  csbwrdg  11101