Theorem sbc8g 2916

Description: This is the closest we can get to df-sbc 2910 if we start from dfsbcq 2911 (see its comments) and dfsbcq2 2912. (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 2911	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		eleq1 2202	. 2 |- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
3		df-clab 2126	. . 3 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		equid 1677	. . . 4 |- y = y
5		dfsbcq2 2912	. . . 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 2747	1 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 1480   [wsb 1735   {cab 2125   [.wsbc 2909

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910

This theorem is referenced by:  bj-elssuniab  12998