Theorem ssrabdv 2126

Description: Subclass of a restricted class abstraction (deduction rule).

Hypotheses

Ref	Expression
ssrabdv.1	|- (ph -> B (_ A)
ssrabdv.2	|- ((ph /\ x e. B) -> ps)

Assertion

Ref	Expression
ssrabdv	|- (ph -> B (_ {x e. A | ps})

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

Proof of Theorem ssrabdv

Step	Hyp	Ref	Expression
1		ssrabdv.1	. . 3 |- (ph -> B (_ A)
2		ssrabdv.2	. . . 4 |- ((ph /\ x e. B) -> ps)
3	2	r19.21aiva 1714	. . 3 |- (ph -> A.x e. B ps)
4	1, 3	jca 288	. 2 |- (ph -> (B (_ A /\ A.x e. B ps))
5		ssrab 2125	. 2 |- (B (_ {x e. A | ps} <-> (B (_ A /\ A.x e. B ps))
6	4, 5	sylibr 200	1 |- (ph -> B (_ {x e. A | ps})

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  A.wral 1645  {crab 1648   (_ wss 2047

This theorem is referenced by:  blss 7853

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rab 1652  df-in 2051  df-ss 2053

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