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Theorem ss2rabi 3229
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 3223 . 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 2528 1  |-  { x  e.  A  |  ph }  C_ 
{ x  e.  A  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141   {crab 2452    C_ wss 3121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-in 3127  df-ss 3134
This theorem is referenced by:  supubti  6972  suplubti  6973
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