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Theorem alimp-no-surprise 38774
 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 38773. The allsome quantifier also counters this problem, see df-alsi 38781. (Contributed by David A. Wheeler, 27-Oct-2018.)
Assertion
Ref Expression
alimp-no-surprise

Proof of Theorem alimp-no-surprise
StepHypRef Expression
1 pm4.82 927 . . . . 5
21albii 1659 . . . 4
3 alnex 1633 . . . 4
42, 3sylbb 197 . . 3
5 imnan 420 . . 3
64, 5mpbi 208 . 2
7 19.26 1699 . . . 4
87anbi2ci 694 . . 3
9 3anass 976 . . 3
10 3anrot 977 . . 3
118, 9, 103bitr2i 273 . 2
126, 11mtbi 296 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 367   w3a 972  wal 1401  wex 1631 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650 This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 974  df-ex 1632 This theorem is referenced by:  alsi-no-surprise  38789
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