Theorem sbc8g

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

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

Proof

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