Theorem csbopab 5008

Description: Move substitution into a class abstraction. Version of csbopabgALT 5009 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 3536	. . . 4 |- ( w = A -> [_ w / x ]_ { <. y , z >. | ph } = [_ A / x ]_ { <. y , z >. | ph } )
2		dfsbcq2 3438	. . . . 5 |- ( w = A -> ( [ w / x ] ph <-> [. A / x ]. ph ) )
3	2	opabbidv 4716	. . . 4 |- ( w = A -> { <. y , z >. | [ w / x ] ph } = { <. y , z >. | [. A / x ]. ph } )
4	1, 3	eqeq12d 2637	. . 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 3203	. . . 4 |- w e. _V
6		nfs1v 2437	. . . . 5 |- F/ x [ w / x ] ph
7	6	nfopab 4718	. . . 4 |- F/_ x { <. y , z >. | [ w / x ] ph }
8		sbequ12 2111	. . . . 5 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
9	8	opabbidv 4716	. . . 4 |- ( x = w -> { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } )
10	5, 7, 9	csbief 3558	. . 3 |- [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph }
11	4, 10	vtoclg 3266	. 2 |- ( A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
12		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = (/) )
13		sbcex 3445	. . . . . . 7 |- ( [. A / x ]. ph -> A e. _V )
14	13	con3i 150	. . . . . 6 |- ( -. A e. _V -> -. [. A / x ]. ph )
15	14	nexdv 1864	. . . . 5 |- ( -. A e. _V -> -. E. z [. A / x ]. ph )
16	15	nexdv 1864	. . . 4 |- ( -. A e. _V -> -. E. y E. z [. A / x ]. ph )
17		opabn0 5006	. . . . 5 |- ( { <. y , z >. | [. A / x ]. ph } =/= (/) <-> E. y E. z [. A / x ]. ph )
18	17	necon1bbii 2843	. . . 4 |- ( -. E. y E. z [. A / x ]. ph <-> { <. y , z >. | [. A / x ]. ph } = (/) )
19	16, 18	sylib 208	. . 3 |- ( -. A e. _V -> { <. y , z >. | [. A / x ]. ph } = (/) )
20	12, 19	eqtr4d 2659	. 2 |- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
21	11, 20	pm2.61i 176	1 |- [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph }

Syntax hints:   -. wn 3   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915   {copab 4712

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ne 2795  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713

This theorem is referenced by:  csbmpt12  5010  csbcnv  5306