Theorem elrabsf 2801

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2696 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 2766	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		elrabsf.1	. . 3 |- F/_ x B
3		nfcv 2178	. . 3 |- F/_ y B
4		nfv 1421	. . 3 |- F/ y ph
5		nfsbc1v 2782	. . 3 |- F/ x [. y / x ]. ph
6		sbceq1a 2773	. . 3 |- ( x = y -> ( ph <-> [. y / x ]. ph ) )
7	2, 3, 4, 5, 6	cbvrab 2555	. 2 |- { x e. B | ph } = { y e. B | [. y / x ]. ph }
8	1, 7	elrab2 2700	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )

Syntax hints:   /\ wa 97   <-> wb 98   e. wcel 1393   F/_wnfc 2165   {crab 2310   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-sbc 2765

This theorem is referenced by:  mpt2xopovel  5856