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Theorem r19.36av 2557
Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if  A has at least one element, but in intuitionistic logic, that is not a sufficient condition. (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 r19.35-1 2556 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  E. x  e.  A  ps )
)
2 idd 21 . . . 4  |-  ( x  e.  A  ->  ( ps  ->  ps ) )
32rexlimiv 2518 . . 3  |-  ( E. x  e.  A  ps  ->  ps )
43imim2i 12 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ps )  ->  ( A. x  e.  A  ph  ->  ps ) )
51, 4syl 14 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 1463   A.wral 2391   E.wrex 2392
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-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-ral 2396  df-rex 2397
This theorem is referenced by:  iinss  3832
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