Theorem ssrab 2121

Description: Subclass of a restricted class abstraction.

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem ssrab

Step	Hyp	Ref	Expression
1		df-rab 1649	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2	1	sseq2i 2082	. 2 |- (B (_ {x e. A | ph} <-> B (_ {x | (x e. A /\ ph)})
3		ssab 2114	. 2 |- (B (_ {x | (x e. A /\ ph)} <-> A.x(x e. B -> (x e. A /\ ph)))
4		dfss3 2055	. . . 4 |- (B (_ A <-> A.x e. B x e. A)
5	4	anbi1i 481	. . 3 |- ((B (_ A /\ A.x e. B ph) <-> (A.x e. B x e. A /\ A.x e. B ph))
6		r19.26 1747	. . 3 |- (A.x e. B (x e. A /\ ph) <-> (A.x e. B x e. A /\ A.x e. B ph))
7		df-ral 1646	. . 3 |- (A.x e. B (x e. A /\ ph) <-> A.x(x e. B -> (x e. A /\ ph)))
8	5, 6, 7	3bitr2r 180	. 2 |- (A.x(x e. B -> (x e. A /\ ph)) <-> (B (_ A /\ A.x e. B ph))
9	2, 3, 8	3bitr 177	1 |- (B (_ {x e. A | ph} <-> (B (_ A /\ A.x e. B ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   e. wcel 956  {cab 1461  A.wral 1642  {crab 1645   (_ wss 2043

This theorem is referenced by:  ssrabdv 2122  efilcp 10481  efilcp2 10486

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rab 1649  df-in 2047  df-ss 2049

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