Theorem ss2rabi 3349

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 3343	. 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 2685	1 |- {x e. A | ph} C_ {x e. A | ps}

Syntax hints:   -> wi 4   e. wcel 1710  {crab 2619   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)