Theorem csbopab 4731

Description: Move substitution into a class abstraction. Version of csbopabgALT 4732 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 3401	. . . 4 |- ( w = A -> [_ w / x ]_ { <. y , z >. | ph } = [_ A / x ]_ { <. y , z >. | ph } )
2		dfsbcq2 3297	. . . . 5 |- ( w = A -> ( [ w / x ] ph <-> [. A / x ]. ph ) )
3	2	opabbidv 4466	. . . 4 |- ( w = A -> { <. y , z >. | [ w / x ] ph } = { <. y , z >. | [. A / x ]. ph } )
4	1, 3	eqeq12d 2476	. . 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 3081	. . . 4 |- w e. _V
6		nfs1v 2151	. . . . 5 |- F/ x [ w / x ] ph
7	6	nfopab 4468	. . . 4 |- F/_ x { <. y , z >. | [ w / x ] ph }
8		sbequ12 1948	. . . . 5 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
9	8	opabbidv 4466	. . . 4 |- ( x = w -> { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } )
10	5, 7, 9	csbief 3423	. . 3 |- [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph }
11	4, 10	vtoclg 3136	. 2 |- ( A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
12		csbprc 3784	. . 3 |- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = (/) )
13		sbcex 3304	. . . . . . 7 |- ( [. A / x ]. ph -> A e. _V )
14	13	con3i 135	. . . . . 6 |- ( -. A e. _V -> -. [. A / x ]. ph )
15	14	nexdv 1823	. . . . 5 |- ( -. A e. _V -> -. E. z [. A / x ]. ph )
16	15	nexdv 1823	. . . 4 |- ( -. A e. _V -> -. E. y E. z [. A / x ]. ph )
17		opabn0 4730	. . . . 5 |- ( { <. y , z >. | [. A / x ]. ph } =/= (/) <-> E. y E. z [. A / x ]. ph )
18	17	necon1bbii 2716	. . . 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 2498	. 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 1370   E.wex 1587   [wsb 1702   e. wcel 1758   _Vcvv 3078   [.wsbc 3294   [_csb 3398   (/)c0 3748   {copab 4460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-opab 4462

This theorem is referenced by:  csbmpt12  4733  csbcnv  5134