Theorem ss2abi 3299

Description: Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Hypothesis

Ref	Expression
ss2abi.1	|- ( ph -> ps )

Assertion

Ref	Expression
ss2abi	|- { x | ph } C_ { x | ps }

Proof of Theorem ss2abi

Step	Hyp	Ref	Expression
1		ss2ab 3295	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )
2		ss2abi.1	. 2 |- ( ph -> ps )
3	1, 2	mpgbir 1501	1 |- { x | ph } C_ { x | ps }

Syntax hints:   -> wi 4   {cab 2217   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213

This theorem is referenced by:  abssi  3302  rabssab  3315  pwsnss  3887  iinuniss  4053  pwpwssunieq  4059  abssexg  4272  imassrn  5087  imadiflem  5409  imainlem  5411  fabexg  5524  f1oabexg  5595  tfrcllemssrecs  6517  mapex  6822  tgval  13344  tgvalex  13345  fngsum  13470  igsumvalx  13471  isghm  13829  wksfval  16172