Theorem ss2abi 3264

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 3260	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )
2		ss2abi.1	. 2 |- ( ph -> ps )
3	1, 2	mpgbir 1475	1 |- { x | ph } C_ { x | ps }

Syntax hints:   -> wi 4   {cab 2190   C_ wss 3165

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-in 3171  df-ss 3178

This theorem is referenced by:  abssi  3267  rabssab  3280  pwsnss  3843  iinuniss  4009  pwpwssunieq  4015  abssexg  4225  imassrn  5032  imadiflem  5352  imainlem  5354  fabexg  5462  f1oabexg  5533  tfrcllemssrecs  6437  mapex  6740  tgval  13065  tgvalex  13066  fngsum  13191  igsumvalx  13192  isghm  13550