Theorem ss2rabi 3330

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 3324	. 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 2608	1 |- { x e. A | ph } C_ { x e. A | ps }

Syntax hints:   -> wi 4   e. wcel 2209   {crab 2532   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-in 3226  df-ss 3233

This theorem is referenced by:  supubti  7329  suplubti  7330  upgruhgr  16266  umgrupgr  16267  umgrislfupgrdom  16286  uspgrushgr  16335  usgruspgr  16338  usgrislfuspgrdom  16345  konigsbergssiedgwen  16641