Theorem ss2rabi 2128

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 2123	. 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 1701	1 |- {x e. A | ph} (_ {x e. A | ps}

Syntax hints:   -> wi 3   e. wcel 958  {crab 1648   (_ wss 2047

This theorem is referenced by:  rankval3 4681  rankval4 4702  fctopOLD 7650  cctop 7652

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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