Theorem sbc8g 2822

Description: This is the closest we can get to df-sbc 2816 if we start from dfsbcq 2817 (see its comments) and dfsbcq2 2818. (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 2817	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		eleq1 2141	. 2 |- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
3		df-clab 2068	. . 3 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		equid 1629	. . . 4 |- y = y
5		dfsbcq2 2818	. . . 4 |- ( y = y -> ( [ y / x ] ph <-> [. y / x ]. ph ) )
6	4, 5	ax-mp 7	. . 3 |- ( [ y / x ] ph <-> [. y / x ]. ph )
7	3, 6	bitr2i 183	. 2 |- ( [. y / x ]. ph <-> y e. { x | ph } )
8	1, 2, 7	vtoclbg 2659	1 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   [wsb 1685   {cab 2067   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  bj-elssuniab  10601