Theorem elrabsf 3474

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3360 has implicit substitution). The hypothesis specifies that x must not be a free variable in B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
elrabsf.1	|- F/_ x B

Assertion

Ref	Expression
elrabsf	|- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )

Proof of Theorem elrabsf

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		elrabsf.1	. . 3 |- F/_ x B
3		nfcv 2764	. . 3 |- F/_ y B
4		nfv 1843	. . 3 |- F/ y ph
5		nfsbc1v 3455	. . 3 |- F/ x [. y / x ]. ph
6		sbceq1a 3446	. . 3 |- ( x = y -> ( ph <-> [. y / x ]. ph ) )
7	2, 3, 4, 5, 6	cbvrab 3198	. 2 |- { x e. B | ph } = { y e. B | [. y / x ]. ph }
8	1, 7	elrab2 3366	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   F/_wnfc 2751   {crab 2916   [.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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  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-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436

This theorem is referenced by:  wfisg  5715  onminesb  6998  mpt2xopovel  7346  ac6num  9301  hashrabsn1  13163  bnj23  30784  bnj1204  31080  tfisg  31716  frinsg  31742  rabrenfdioph  37378