Theorem csbabg 2901

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 2827	. . . 4 |- ( [.z / y ]. [.A / x ].ph <-> [.A / x ]. [.z / y ].ph)
2		df-clab 2024	. . . . 5 |- (z e. {y | [.A / x ].ph } <-> [z / y] [.A / x ].ph)
3		sbsbc 2762	. . . . 5 |- ([z / y] [.A / x ].ph <-> [.z / y ]. [.A / x ].ph)
4	2, 3	bitri 173	. . . 4 |- (z e. {y | [.A / x ].ph } <-> [.z / y ]. [.A / x ].ph)
5		df-clab 2024	. . . . . 6 |- (z e. {y | ph } <-> [z / y]ph)
6		sbsbc 2762	. . . . . 6 |- ([z / y]ph <-> [.z / y ].ph)
7	5, 6	bitri 173	. . . . 5 |- (z e. {y | ph } <-> [.z / y ].ph)
8	7	sbcbii 2812	. . . 4 |- ( [.A / x ].z e. {y | ph } <-> [.A / x ]. [.z / y ].ph)
9	1, 4, 8	3bitr4i 201	. . 3 |- (z e. {y | [.A / x ].ph } <-> [.A / x ].z e. {y | ph })
10		sbcel2g 2865	. . 3 |- (A e. V -> ( [.A / x ].z e. {y | ph } <-> z e. [_A / x ]_ {y | ph }))
11	9, 10	syl5rbb 182	. 2 |- (A e. V -> (z e. [_A / x ]_ {y | ph } <-> z e. {y | [.A / x ].ph }))
12	11	eqrdv 2035	1 |- (A e. V -> [_A / x ]_ {y | ph } = {y | [.A / x ].ph })

Syntax hints:   -> wi 4   = wceq 1242   e. wcel 1390  [wsb 1642   {cab 2023   [.wsbc 2758   [_csb 2846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847

This theorem is referenced by:  csbsng  3422  csbunig  3579  csbxpg  4364  csbdmg  4472  csbrng  4725