Theorem ss2abi 3273

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 3269	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )
2		ss2abi.1	. 2 |- ( ph -> ps )
3	1, 2	mpgbir 1477	1 |- { x | ph } C_ { x | ps }

Syntax hints:   -> wi 4   {cab 2193   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187

This theorem is referenced by:  abssi  3276  rabssab  3289  pwsnss  3858  iinuniss  4024  pwpwssunieq  4030  abssexg  4242  imassrn  5052  imadiflem  5372  imainlem  5374  fabexg  5485  f1oabexg  5556  tfrcllemssrecs  6461  mapex  6764  tgval  13209  tgvalex  13210  fngsum  13335  igsumvalx  13336  isghm  13694