Theorem ss2rabdv 2127

Description: Deduction of restricted abstraction subclass from implication.

Hypothesis

Ref	Expression
ss2rabdv.1	|- ((ph /\ x e. A) -> (ps -> ch))

Assertion

Ref	Expression
ss2rabdv	|- (ph -> {x e. A | ps} (_ {x e. A | ch})

Distinct variable group:   ph,x

Proof of Theorem ss2rabdv

Step	Hyp	Ref	Expression
1		ss2rabdv.1	. . 3 |- ((ph /\ x e. A) -> (ps -> ch))
2	1	r19.21aiva 1714	. 2 |- (ph -> A.x e. A (ps -> ch))
3		ss2rab 2123	. 2 |- ({x e. A | ps} (_ {x e. A | ch} <-> A.x e. A (ps -> ch))
4	2, 3	sylibr 200	1 |- (ph -> {x e. A | ps} (_ {x e. A | ch})

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

This theorem is referenced by:  rankr1id 4697  iooss1 6373  iooss2 6374  fzss1t 6503  fzss2t 6504  clsss 7687  pjspansnt 9500

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-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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