Theorem ss2abi 3252

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 3248	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )
2		ss2abi.1	. 2 |- ( ph -> ps )
3	1, 2	mpgbir 1464	1 |- { x | ph } C_ { x | ps }

Syntax hints:   -> wi 4   {cab 2179   C_ wss 3154

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3160  df-ss 3167

This theorem is referenced by:  abssi  3255  rabssab  3268  pwsnss  3830  iinuniss  3996  pwpwssunieq  4002  abssexg  4212  imassrn  5017  imadiflem  5334  imainlem  5336  fabexg  5442  f1oabexg  5513  tfrcllemssrecs  6407  mapex  6710  tgval  12876  tgvalex  12877  fngsum  12974  igsumvalx  12975  isghm  13316