Theorem elssabg 4819

Description: Membership in a class abstraction involving a subset. Unlike elabg 3351, 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 4804	. . . 4 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
2	1	expcom 451	. . 3 |- ( B e. V -> ( A C_ B -> A e. _V ) )
3	2	adantrd 484	. 2 |- ( B e. V -> ( ( A C_ B /\ ps ) -> A e. _V ) )
4		sseq1 3626	. . . 4 |- ( x = A -> ( x C_ B <-> A C_ B ) )
5		elssabg.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
6	4, 5	anbi12d 747	. . 3 |- ( x = A -> ( ( x C_ B /\ ph ) <-> ( A C_ B /\ ps ) ) )
7	6	elab3g 3357	. 2 |- ( ( ( A C_ B /\ ps ) -> A e. _V ) -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )
8	3, 7	syl 17	1 |- ( B e. V -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574

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  ax-sep 4781

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-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by: (None)