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Theorem alimp-no-surprise 31485
Description: There is no "surprise" in a for-all with implication if there exists a value where the antecedent is true. This is one way to prevent for-all with implication from allowing anything. For a contrast, see alimp-surprise 31484. The allsome quantifier also counters this problem, see df-alsi 31492. (Contributed by David A. Wheeler, 27-Oct-2018.)
Assertion
Ref Expression
alimp-no-surprise  |-  -.  ( A. x ( ph  ->  ps )  /\  A. x
( ph  ->  -.  ps )  /\  E. x ph )

Proof of Theorem alimp-no-surprise
StepHypRef Expression
1 pm4.82 919 . . . . 5  |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  -. 
ps ) )  <->  -.  ph )
21albii 1611 . . . 4  |-  ( A. x ( ( ph  ->  ps )  /\  ( ph  ->  -.  ps )
)  <->  A. x  -.  ph )
3 alnex 1589 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
42, 3sylbb 197 . . 3  |-  ( A. x ( ( ph  ->  ps )  /\  ( ph  ->  -.  ps )
)  ->  -.  E. x ph )
5 imnan 422 . . 3  |-  ( ( A. x ( (
ph  ->  ps )  /\  ( ph  ->  -.  ps )
)  ->  -.  E. x ph )  <->  -.  ( A. x ( ( ph  ->  ps )  /\  ( ph  ->  -.  ps )
)  /\  E. x ph ) )
64, 5mpbi 208 . 2  |-  -.  ( A. x ( ( ph  ->  ps )  /\  ( ph  ->  -.  ps )
)  /\  E. x ph )
7 19.26 1648 . . . 4  |-  ( A. x ( ( ph  ->  ps )  /\  ( ph  ->  -.  ps )
)  <->  ( A. x
( ph  ->  ps )  /\  A. x ( ph  ->  -.  ps ) ) )
87anbi2ci 696 . . 3  |-  ( ( A. x ( (
ph  ->  ps )  /\  ( ph  ->  -.  ps )
)  /\  E. x ph )  <->  ( E. x ph  /\  ( A. x
( ph  ->  ps )  /\  A. x ( ph  ->  -.  ps ) ) ) )
9 3anass 969 . . 3  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps )  /\  A. x
( ph  ->  -.  ps ) )  <->  ( E. x ph  /\  ( A. x ( ph  ->  ps )  /\  A. x
( ph  ->  -.  ps ) ) ) )
10 3anrot 970 . . 3  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps )  /\  A. x
( ph  ->  -.  ps ) )  <->  ( A. x ( ph  ->  ps )  /\  A. x
( ph  ->  -.  ps )  /\  E. x ph ) )
118, 9, 103bitr2i 273 . 2  |-  ( ( A. x ( (
ph  ->  ps )  /\  ( ph  ->  -.  ps )
)  /\  E. x ph )  <->  ( A. x
( ph  ->  ps )  /\  A. x ( ph  ->  -.  ps )  /\  E. x ph ) )
126, 11mtbi 298 1  |-  -.  ( A. x ( ph  ->  ps )  /\  A. x
( ph  ->  -.  ps )  /\  E. x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1368   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-ex 1588
This theorem is referenced by:  alsi-no-surprise  31500
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