Theorem csbab 4008

Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 19-Aug-2018.)

Assertion

Ref	Expression
csbab	|- [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph }

Distinct variable groups:   y, A   x, y

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

Proof of Theorem csbab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clab 2609	. . . 4 |- ( z e. { y | [. A / x ]. ph } <-> [ z / y ] [. A / x ]. ph )
2		sbsbc 3439	. . . 4 |- ( [ z / y ] [. A / x ]. ph <-> [. z / y ]. [. A / x ]. ph )
3	1, 2	bitri 264	. . 3 |- ( z e. { y | [. A / x ]. ph } <-> [. z / y ]. [. A / x ]. ph )
4		sbccom 3509	. . . 4 |- ( [. z / y ]. [. A / x ]. ph <-> [. A / x ]. [. z / y ]. 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	4, 8	bitr4i 267	. . 3 |- ( [. z / y ]. [. A / x ]. ph <-> [. A / x ]. z e. { y | ph } )
10		sbcel2 3989	. . 3 |- ( [. A / x ]. z e. { y | ph } <-> z e. [_ A / x ]_ { y | ph } )
11	3, 9, 10	3bitrri 287	. 2 |- ( z e. [_ A / x ]_ { y | ph } <-> z e. { y | [. A / x ]. ph } )
12	11	eqriv 2619	1 |- [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph }

Syntax hints:   = 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-fal 1489  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  df-dif 3577  df-nul 3916

This theorem is referenced by:  csbsng  4243  csbuni  4466  csbxp  5200  csbdm  5318  csbwrdg  13334  abfmpeld  29454  abfmpel  29455  csbwrecsg  33173  csboprabg  33176  csbfinxpg  33225  csbfv12gALTOLD  39052  csbfv12gALTVD  39135