Theorem ss2rabi 3076

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 3070	. 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 2421	1 |- { x e. A | ph } C_ { x e. A | ps }

Syntax hints:   -> wi 4   e. wcel 1433   {crab 2352   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-in 2979  df-ss 2986

This theorem is referenced by:  supubti  6412  suplubti  6413