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Theorem r19.36av 2816
Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when  A is empty. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
r19.36av  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  ps ) )
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem r19.36av
StepHypRef Expression
1 r19.35 2815 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
2 idd 22 . . . 4  |-  ( x  e.  A  ->  ( ps  ->  ps ) )
32rexlimiv 2784 . . 3  |-  ( E. x  e.  A  ps  ->  ps )
43imim2i 14 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ps )  ->  ( A. x  e.  A  ph  ->  ps ) )
51, 4sylbi 188 1  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   A.wral 2666   E.wrex 2667
This theorem is referenced by:  iinss  4102  uniimadom  8375  hashgt12el  11637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-ral 2671  df-rex 2672
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