Theorem bnj1454 30912

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

Hypothesis

Ref	Expression
bnj1454.1	|- A = { x | ph }

Assertion

Ref	Expression
bnj1454	|- ( B e. _V -> ( B e. A <-> [. B / x ]. ph ) )

Proof of Theorem bnj1454

Step	Hyp	Ref	Expression
1		df-sbc 3436	. . 3 |- ( [. B / x ]. ph <-> B e. { x | ph } )
2	1	a1i 11	. 2 |- ( B e. _V -> ( [. B / x ]. ph <-> B e. { x | ph } ) )
3		bnj1454.1	. . 3 |- A = { x | ph }
4	3	eleq2i 2693	. 2 |- ( B e. A <-> B e. { x | ph } )
5	2, 4	syl6rbbr 279	1 |- ( B e. _V -> ( B e. A <-> [. B / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   [.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-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  bnj1452  31120  bnj1463  31123