Theorem bnj1454

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj1454.1	|- A = { x | ph }
Assertion	bnj1454	|- ( B e. _V -> ( B e. A <-> [. B / x ]. ph ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1454.1	|- A = { x | ph }
2	1	eleq2i	|- ( B e. A <-> B e. { x | ph } )
3		df-sbc	|- ( [. B / x ]. ph <-> B e. { x | ph } )
4	3	a1i	|- ( B e. _V -> ( [. B / x ]. ph <-> B e. { x | ph } ) )
5	2 4	bitr4id	|- ( B e. _V -> ( B e. A <-> [. B / x ]. ph ) )