Theorem csboprabg 33176

Description: Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)

Assertion

Ref	Expression
csboprabg	|- ( A e. V -> [_ A / x ]_ { <. <. y , z >. , d >. | ph } = { <. <. y , z >. , d >. | [. A / x ]. ph } )

Distinct variable groups:   A, d   y, A   z, A   V, d   y, V   z, V   x, d   x, y   x, z

Allowed substitution hints:   ph( x, y, z, d)   A( x)   V( x)

Proof of Theorem csboprabg

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbab 4008	. . 3 |- [_ A / x ]_ { c | E. y E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) } = { c | [. A / x ]. E. y E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) }
2		sbcex2 3486	. . . . 5 |- ( [. A / x ]. E. y E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) <-> E. y [. A / x ]. E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) )
3		sbcex2 3486	. . . . . . 7 |- ( [. A / x ]. E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) <-> E. z [. A / x ]. E. d ( c = <. <. y , z >. , d >. /\ ph ) )
4		sbcex2 3486	. . . . . . . . 9 |- ( [. A / x ]. E. d ( c = <. <. y , z >. , d >. /\ ph ) <-> E. d [. A / x ]. ( c = <. <. y , z >. , d >. /\ ph ) )
5		sbcan 3478	. . . . . . . . . . 11 |- ( [. A / x ]. ( c = <. <. y , z >. , d >. /\ ph ) <-> ( [. A / x ]. c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) )
6		sbcg 3503	. . . . . . . . . . . 12 |- ( A e. V -> ( [. A / x ]. c = <. <. y , z >. , d >. <-> c = <. <. y , z >. , d >. ) )
7	6	anbi1d 741	. . . . . . . . . . 11 |- ( A e. V -> ( ( [. A / x ]. c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) <-> ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) ) )
8	5, 7	syl5bb 272	. . . . . . . . . 10 |- ( A e. V -> ( [. A / x ]. ( c = <. <. y , z >. , d >. /\ ph ) <-> ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) ) )
9	8	exbidv 1850	. . . . . . . . 9 |- ( A e. V -> ( E. d [. A / x ]. ( c = <. <. y , z >. , d >. /\ ph ) <-> E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) ) )
10	4, 9	syl5bb 272	. . . . . . . 8 |- ( A e. V -> ( [. A / x ]. E. d ( c = <. <. y , z >. , d >. /\ ph ) <-> E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) ) )
11	10	exbidv 1850	. . . . . . 7 |- ( A e. V -> ( E. z [. A / x ]. E. d ( c = <. <. y , z >. , d >. /\ ph ) <-> E. z E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) ) )
12	3, 11	syl5bb 272	. . . . . 6 |- ( A e. V -> ( [. A / x ]. E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) <-> E. z E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) ) )
13	12	exbidv 1850	. . . . 5 |- ( A e. V -> ( E. y [. A / x ]. E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) <-> E. y E. z E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) ) )
14	2, 13	syl5bb 272	. . . 4 |- ( A e. V -> ( [. A / x ]. E. y E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) <-> E. y E. z E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) ) )
15	14	abbidv 2741	. . 3 |- ( A e. V -> { c | [. A / x ]. E. y E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) } = { c | E. y E. z E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) } )
16	1, 15	syl5eq 2668	. 2 |- ( A e. V -> [_ A / x ]_ { c | E. y E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) } = { c | E. y E. z E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) } )
17		df-oprab 6654	. . 3 |- { <. <. y , z >. , d >. | ph } = { c | E. y E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) }
18	17	csbeq2i 3993	. 2 |- [_ A / x ]_ { <. <. y , z >. , d >. | ph } = [_ A / x ]_ { c | E. y E. z E. d ( c = <. <. y , z >. , d >. /\ ph ) }
19		df-oprab 6654	. 2 |- { <. <. y , z >. , d >. | [. A / x ]. ph } = { c | E. y E. z E. d ( c = <. <. y , z >. , d >. /\ [. A / x ]. ph ) }
20	16, 18, 19	3eqtr4g 2681	1 |- ( A e. V -> [_ A / x ]_ { <. <. y , z >. , d >. | ph } = { <. <. y , z >. , d >. | [. A / x ]. ph } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533   <.cop 4183   {coprab 6651

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  df-oprab 6654

This theorem is referenced by:  csbmpt22g  33177