Theorem csbopabg 4111

Description: Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
csbopabg	|- ( A e. V -> [_ 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)   V( x, y, z)

Proof of Theorem csbopabg

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3087	. . 3 |- ( w = A -> [_ w / x ]_ { <. y , z >. | ph } = [_ A / x ]_ { <. y , z >. | ph } )
2		dfsbcq2 2992	. . . 4 |- ( w = A -> ( [ w / x ] ph <-> [. A / x ]. ph ) )
3	2	opabbidv 4099	. . 3 |- ( w = A -> { <. y , z >. | [ w / x ] ph } = { <. y , z >. | [. A / x ]. ph } )
4	1, 3	eqeq12d 2211	. 2 |- ( w = A -> ( [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } <-> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } ) )
5		vex 2766	. . 3 |- w e. _V
6		nfs1v 1958	. . . 4 |- F/ x [ w / x ] ph
7	6	nfopab 4101	. . 3 |- F/_ x { <. y , z >. | [ w / x ] ph }
8		sbequ12 1785	. . . 4 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
9	8	opabbidv 4099	. . 3 |- ( x = w -> { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } )
10	5, 7, 9	csbief 3129	. 2 |- [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph }
11	4, 10	vtoclg 2824	1 |- ( A e. V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )

Syntax hints:   -> wi 4   = wceq 1364   [wsb 1776   e. wcel 2167   [.wsbc 2989   [_csb 3084   {copab 4093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990  df-csb 3085  df-opab 4095

This theorem is referenced by:  csbcnvg  4850