Theorem ss2rabi 2124

Description: Inference of restricted abstraction subclass from implication.

Hypothesis

Ref	Expression
ss2rabi.1	|- (x e. A -> (ph -> ps))

Assertion

Ref	Expression
ss2rabi	|- {x e. A | ph} (_ {x e. A | ps}

Proof of Theorem ss2rabi

Step	Hyp	Ref	Expression
1		ss2rab 2119	. 2 |- ({x e. A | ph} (_ {x e. A | ps} <-> A.x e. A (ph -> ps))
2		ss2rabi.1	. 2 |- (x e. A -> (ph -> ps))
3	1, 2	mprgbir 1698	1 |- {x e. A | ph} (_ {x e. A | ps}

Syntax hints:   -> wi 3   e. wcel 956  {crab 1645   (_ wss 2043

This theorem is referenced by:  rankval3 4661  rankval4 4682  fctop 7600  cctop 7602

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-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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