Theorem sbc8g 2831

Description: This is the closest we can get to df-sbc 2825 if we start from dfsbcq 2826 (see its comments) and dfsbcq2 2827. (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 2826	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		eleq1 2145	. 2 |- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
3		df-clab 2070	. . 3 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		equid 1630	. . . 4 |- y = y
5		dfsbcq2 2827	. . . 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 2668	1 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1434   [wsb 1687   {cab 2069   [.wsbc 2824

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825

This theorem is referenced by:  bj-elssuniab  10861