Theorem sbc8g 3443

Description: This is the closest we can get to df-sbc 3436 if we start from dfsbcq 3437 (see its comments) and dfsbcq2 3438. (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 3437	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		eleq1 2689	. 2 |- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
3		df-clab 2609	. . 3 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		equid 1939	. . . 4 |- y = y
5		dfsbcq2 3438	. . . 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 265	. 2 |- ( [. y / x ]. ph <-> y e. { x | ph } )
8	1, 2, 7	vtoclbg 3267	1 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 196   [wsb 1880   e. wcel 1990   {cab 2608   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  bnj984  31022  rusbcALT  38640