Theorem ss2rabi 3022

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 3016	. 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 2379	1 |- { x e. A | ph } C_ { x e. A | ps }

Syntax hints:   -> wi 4   e. wcel 1393   {crab 2310   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-in 2924  df-ss 2931

This theorem is referenced by: (None)