Theorem csbopab 4733

Description: Move substitution into a class abstraction. Version of csbopabgALT 4734 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 3352	. . . 4 |- ( w = A -> [_ w / x ]_ { <. y , z >. | ph } = [_ A / x ]_ { <. y , z >. | ph } )
2		dfsbcq2 3258	. . . . 5 |- ( w = A -> ( [ w / x ] ph <-> [. A / x ]. ph ) )
3	2	opabbidv 4459	. . . 4 |- ( w = A -> { <. y , z >. | [ w / x ] ph } = { <. y , z >. | [. A / x ]. ph } )
4	1, 3	eqeq12d 2486	. . 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 3034	. . . 4 |- w e. _V
6		nfs1v 2286	. . . . 5 |- F/ x [ w / x ] ph
7	6	nfopab 4461	. . . 4 |- F/_ x { <. y , z >. | [ w / x ] ph }
8		sbequ12 2098	. . . . 5 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
9	8	opabbidv 4459	. . . 4 |- ( x = w -> { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } )
10	5, 7, 9	csbief 3374	. . 3 |- [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph }
11	4, 10	vtoclg 3093	. 2 |- ( A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
12		csbprc 3774	. . 3 |- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = (/) )
13		sbcex 3265	. . . . . . 7 |- ( [. A / x ]. ph -> A e. _V )
14	13	con3i 142	. . . . . 6 |- ( -. A e. _V -> -. [. A / x ]. ph )
15	14	nexdv 1790	. . . . 5 |- ( -. A e. _V -> -. E. z [. A / x ]. ph )
16	15	nexdv 1790	. . . 4 |- ( -. A e. _V -> -. E. y E. z [. A / x ]. ph )
17		opabn0 4732	. . . . 5 |- ( { <. y , z >. | [. A / x ]. ph } =/= (/) <-> E. y E. z [. A / x ]. ph )
18	17	necon1bbii 2692	. . . 4 |- ( -. E. y E. z [. A / x ]. ph <-> { <. y , z >. | [. A / x ]. ph } = (/) )
19	16, 18	sylib 201	. . 3 |- ( -. A e. _V -> { <. y , z >. | [. A / x ]. ph } = (/) )
20	12, 19	eqtr4d 2508	. 2 |- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
21	11, 20	pm2.61i 169	1 |- [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph }

Syntax hints:   -. wn 3   = wceq 1452   E.wex 1671   [wsb 1805   e. wcel 1904   _Vcvv 3031   [.wsbc 3255   [_csb 3349   (/)c0 3722   {copab 4453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455

This theorem is referenced by:  csbmpt12  4735  csbcnv  5023