Theorem sbc8g 2958

Description: This is the closest we can get to df-sbc 2952 if we start from dfsbcq 2953 (see its comments) and dfsbcq2 2954. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbc8g	|- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )

Proof of Theorem sbc8g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 2953	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		eleq1 2229	. 2 |- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
3		df-clab 2152	. . 3 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		equid 1689	. . . 4 |- y = y
5		dfsbcq2 2954	. . . 4 |- ( y = y -> ( [ y / x ] ph <-> [. y / x ]. ph ) )
6	4, 5	ax-mp 5	. . 3 |- ( [ y / x ] ph <-> [. y / x ]. ph )
7	3, 6	bitr2i 184	. 2 |- ( [. y / x ]. ph <-> y e. { x | ph } )
8	1, 2, 7	vtoclbg 2787	1 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 104   [wsb 1750   e. wcel 2136   {cab 2151   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by:  bj-elssuniab  13672