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Theorem ss2rabi 3249
Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
Hypothesis
Ref Expression
ss2rabi.1  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
ss2rabi  |-  { x  e.  A  |  ph }  C_ 
{ x  e.  A  |  ps }

Proof of Theorem ss2rabi
StepHypRef Expression
1 ss2rab 3243 . 2  |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps } 
<-> 
A. x  e.  A  ( ph  ->  ps )
)
2 ss2rabi.1 . 2  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
31, 2mprgbir 2545 1  |-  { x  e.  A  |  ph }  C_ 
{ x  e.  A  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2158   {crab 2469    C_ wss 3141
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rab 2474  df-in 3147  df-ss 3154
This theorem is referenced by:  supubti  7011  suplubti  7012
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