ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.36av Unicode version

Theorem r19.36av 2461
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 2460 . 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 2427 . . 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 1393   A.wral 2306   E.wrex 2307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312
This theorem is referenced by:  iinss  3708
  Copyright terms: Public domain W3C validator