Theorem elssabg 2726

Description: Membership in a class abstraction involving a subset. Unlike elabg 1899, A does not have to be a set.

Hypothesis

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

Assertion

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

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

Proof of Theorem elssabg

Step	Hyp	Ref	Expression
1		ssexg 2721	. . . 4 |- ((A (_ B /\ B e. C) -> A e. V)
2	1	expcom 374	. . 3 |- (B e. C -> (A (_ B -> A e. V))
3	2	adantrd 391	. 2 |- (B e. C -> ((A (_ B /\ ps) -> A e. V))
4		sseq1 2082	. . . 4 |- (x = A -> (x (_ B <-> A (_ B))
5		elssabg.1	. . . 4 |- (x = A -> (ph <-> ps))
6	4, 5	anbi12d 628	. . 3 |- (x = A -> ((x (_ B /\ ph) <-> (A (_ B /\ ps)))
7	6	elab3g 1902	. 2 |- (((A (_ B /\ ps) -> A e. V) -> (A e. {x | (x (_ B /\ ph)} <-> (A (_ B /\ ps)))
8	3, 7	syl 10	1 |- (B e. C -> (A e. {x | (x (_ B /\ ph)} <-> (A (_ B /\ ps)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   (_ wss 2047

This theorem is referenced by:  isopn 7859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053

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