Theorem ss2rabi 3275

Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

Hypothesis

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

Assertion

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

Proof of Theorem ss2rabi

Step	Hyp	Ref	Expression
1		ss2rab 3269	. 2 |- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )
2		ss2rabi.1	. 2 |- ( x e. A -> ( ph -> ps ) )
3	1, 2	mprgbir 2564	1 |- { x e. A | ph } C_ { x e. A | ps }

Syntax hints:   -> wi 4   e. wcel 2176   {crab 2488   C_ wss 3166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-in 3172  df-ss 3179

This theorem is referenced by:  supubti  7101  suplubti  7102