Theorem ss2rabi 3224

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 3218	. 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 2524	1 |- { x e. A | ph } C_ { x e. A | ps }

Syntax hints:   -> wi 4   e. wcel 2136   {crab 2448   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-in 3122  df-ss 3129

This theorem is referenced by:  supubti  6964  suplubti  6965