Theorem csbopab 4788

Description: Move substitution into a class abstraction. Version of csbopabgALT 4789 without a sethood antecedent but depending on more axioms. (Contributed by NM, 6-Aug-2007.) (Revised by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbopab	|- [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph }

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

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

Proof of Theorem csbopab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3433	. . . 4 |- ( w = A -> [_ w / x ]_ { <. y , z >. | ph } = [_ A / x ]_ { <. y , z >. | ph } )
2		dfsbcq2 3330	. . . . 5 |- ( w = A -> ( [ w / x ] ph <-> [. A / x ]. ph ) )
3	2	opabbidv 4520	. . . 4 |- ( w = A -> { <. y , z >. | [ w / x ] ph } = { <. y , z >. | [. A / x ]. ph } )
4	1, 3	eqeq12d 2479	. . 3 |- ( w = A -> ( [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } <-> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } ) )
5		vex 3112	. . . 4 |- w e. _V
6		nfs1v 2182	. . . . 5 |- F/ x [ w / x ] ph
7	6	nfopab 4522	. . . 4 |- F/_ x { <. y , z >. | [ w / x ] ph }
8		sbequ12 1993	. . . . 5 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
9	8	opabbidv 4520	. . . 4 |- ( x = w -> { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } )
10	5, 7, 9	csbief 3455	. . 3 |- [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph }
11	4, 10	vtoclg 3167	. 2 |- ( A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
12		csbprc 3830	. . 3 |- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = (/) )
13		sbcex 3337	. . . . . . 7 |- ( [. A / x ]. ph -> A e. _V )
14	13	con3i 135	. . . . . 6 |- ( -. A e. _V -> -. [. A / x ]. ph )
15	14	nexdv 1885	. . . . 5 |- ( -. A e. _V -> -. E. z [. A / x ]. ph )
16	15	nexdv 1885	. . . 4 |- ( -. A e. _V -> -. E. y E. z [. A / x ]. ph )
17		opabn0 4787	. . . . 5 |- ( { <. y , z >. | [. A / x ]. ph } =/= (/) <-> E. y E. z [. A / x ]. ph )
18	17	necon1bbii 2721	. . . 4 |- ( -. E. y E. z [. A / x ]. ph <-> { <. y , z >. | [. A / x ]. ph } = (/) )
19	16, 18	sylib 196	. . 3 |- ( -. A e. _V -> { <. y , z >. | [. A / x ]. ph } = (/) )
20	12, 19	eqtr4d 2501	. 2 |- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
21	11, 20	pm2.61i 164	1 |- [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph }

Syntax hints:   -. wn 3   = wceq 1395   E.wex 1613   [wsb 1740   e. wcel 1819   _Vcvv 3109   [.wsbc 3327   [_csb 3430   (/)c0 3793   {copab 4514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516

This theorem is referenced by:  csbmpt12  4790  csbcnv  5196