Theorem ss2abi 3310

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 3306	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )
2		ss2abi.1	. 2 |- ( ph -> ps )
3	1, 2	mpgbir 1502	1 |- { x | ph } C_ { x | ps }

Syntax hints:   -> wi 4   {cab 2218   C_ wss 3211

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224

This theorem is referenced by:  abssi  3313  rabssab  3327  pwsnss  3908  iinuniss  4074  pwpwssunieq  4080  abssexg  4295  imassrn  5112  imadiflem  5435  imainlem  5437  fabexg  5554  f1oabexg  5626  tfrcllemssrecs  6583  mapex  6888  ballotfilem2  13142  tgval  13475  tgvalex  13476  fngsum  13601  igsumvalx  13602  isghm  13960  wksfval  16317