Theorem csbopabg 4122

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 3096	. . 3 |- ( w = A -> [_ w / x ]_ { <. y , z >. | ph } = [_ A / x ]_ { <. y , z >. | ph } )
2		dfsbcq2 3001	. . . 4 |- ( w = A -> ( [ w / x ] ph <-> [. A / x ]. ph ) )
3	2	opabbidv 4110	. . 3 |- ( w = A -> { <. y , z >. | [ w / x ] ph } = { <. y , z >. | [. A / x ]. ph } )
4	1, 3	eqeq12d 2220	. 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 2775	. . 3 |- w e. _V
6		nfs1v 1967	. . . 4 |- F/ x [ w / x ] ph
7	6	nfopab 4112	. . 3 |- F/_ x { <. y , z >. | [ w / x ] ph }
8		sbequ12 1794	. . . 4 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
9	8	opabbidv 4110	. . 3 |- ( x = w -> { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } )
10	5, 7, 9	csbief 3138	. 2 |- [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph }
11	4, 10	vtoclg 2833	1 |- ( A e. V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )

Syntax hints:   -> wi 4   = wceq 1373   [wsb 1785   e. wcel 2176   [.wsbc 2998   [_csb 3093   {copab 4104

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999  df-csb 3094  df-opab 4106

This theorem is referenced by:  csbcnvg  4862