Theorem ss2abi 3297

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 3293	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )
2		ss2abi.1	. 2 |- ( ph -> ps )
3	1, 2	mpgbir 1499	1 |- { x | ph } C_ { x | ps }

Syntax hints:   -> wi 4   {cab 2215   C_ wss 3198

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3204  df-ss 3211

This theorem is referenced by:  abssi  3300  rabssab  3313  pwsnss  3885  iinuniss  4051  pwpwssunieq  4057  abssexg  4270  imassrn  5085  imadiflem  5406  imainlem  5408  fabexg  5521  f1oabexg  5592  tfrcllemssrecs  6513  mapex  6818  tgval  13335  tgvalex  13336  fngsum  13461  igsumvalx  13462  isghm  13820  wksfval  16119