Theorem ss2abi 3296

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 3292	. 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 3197

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 3203  df-ss 3210

This theorem is referenced by:  abssi  3299  rabssab  3312  pwsnss  3881  iinuniss  4047  pwpwssunieq  4053  abssexg  4265  imassrn  5078  imadiflem  5399  imainlem  5401  fabexg  5512  f1oabexg  5583  tfrcllemssrecs  6496  mapex  6799  tgval  13290  tgvalex  13291  fngsum  13416  igsumvalx  13417  isghm  13775  wksfval  16028