Theorem elssabg 4013

Description: Membership in a class abstraction involving a subset. Unlike elabg 2783, A does not have to be a set. (Contributed by NM, 29-Aug-2006.)

Hypothesis

Ref	Expression
elssabg.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
elssabg	|- ( B e. V -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem elssabg

Step	Hyp	Ref	Expression
1		ssexg 4007	. . . 4 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
2	1	expcom 115	. . 3 |- ( B e. V -> ( A C_ B -> A e. _V ) )
3	2	adantrd 275	. 2 |- ( B e. V -> ( ( A C_ B /\ ps ) -> A e. _V ) )
4		sseq1 3070	. . . 4 |- ( x = A -> ( x C_ B <-> A C_ B ) )
5		elssabg.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
6	4, 5	anbi12d 460	. . 3 |- ( x = A -> ( ( x C_ B /\ ph ) <-> ( A C_ B /\ ps ) ) )
7	6	elab3g 2788	. 2 |- ( ( ( A C_ B /\ ps ) -> A e. _V ) -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )
8	3, 7	syl 14	1 |- ( B e. V -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   {cab 2086   _Vcvv 2641   C_ wss 3021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034

This theorem is referenced by: (None)