Theorem csbabgOLD 39050

Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 19-Aug-2018. Use csbab 4008 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbabgOLD	|- ( 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 csbabgOLD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbccom 3509	. . . 4 |- ( [. z / y ]. [. A / x ]. ph <-> [. A / x ]. [. z / y ]. ph )
2		df-clab 2609	. . . . 5 |- ( z e. { y | [. A / x ]. ph } <-> [ z / y ] [. A / x ]. ph )
3		sbsbc 3439	. . . . 5 |- ( [ z / y ] [. A / x ]. ph <-> [. z / y ]. [. A / x ]. ph )
4	2, 3	bitri 264	. . . 4 |- ( z e. { y | [. A / x ]. ph } <-> [. z / y ]. [. A / x ]. ph )
5		df-clab 2609	. . . . . 6 |- ( z e. { y | ph } <-> [ z / y ] ph )
6		sbsbc 3439	. . . . . 6 |- ( [ z / y ] ph <-> [. z / y ]. ph )
7	5, 6	bitri 264	. . . . 5 |- ( z e. { y | ph } <-> [. z / y ]. ph )
8	7	sbcbii 3491	. . . 4 |- ( [. A / x ]. z e. { y | ph } <-> [. A / x ]. [. z / y ]. ph )
9	1, 4, 8	3bitr4i 292	. . 3 |- ( z e. { y | [. A / x ]. ph } <-> [. A / x ]. z e. { y | ph } )
10		sbcel2gOLD 38755	. . 3 |- ( A e. V -> ( [. A / x ]. z e. { y | ph } <-> z e. [_ A / x ]_ { y | ph } ) )
11	9, 10	syl5rbb 273	. 2 |- ( A e. V -> ( z e. [_ A / x ]_ { y | ph } <-> z e. { y | [. A / x ]. ph } ) )
12	11	eqrdv 2620	1 |- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )

Syntax hints:   -> wi 4   = wceq 1483   [wsb 1880   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbunigOLD  39051  csbxpgOLD  39053  csbrngOLD  39056  csbingVD  39120  csbsngVD  39129  csbxpgVD  39130  csbrngVD  39132  csbunigVD  39134