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Theorem eximp-surprise 43054
Description: Show what implication inside "there exists" really expands to (using implication directly inside "there exists" is usually a mistake).

Those inexperienced with formal notations of classical logic may use expressions combining "there exists" with implication. That is usually a mistake, because as proven using imor 842, such an expression can be rewritten using not with or - and that is often not what the author intended. New users of formal notation who use "there exists" with an implication should consider if they meant "and" instead of "implies". A stark example is shown in eximp-surprise2 43055. See also alimp-surprise 43050 and empty-surprise 43052. (Contributed by David A. Wheeler, 17-Oct-2018.)

Ref Expression
eximp-surprise (∃𝑥(𝜑𝜓) ↔ ∃𝑥𝜑𝜓))

Proof of Theorem eximp-surprise
StepHypRef Expression
1 imor 842 . 2 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
21exbii 1924 1 (∃𝑥(𝜑𝜓) ↔ ∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 836  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885
This theorem depends on definitions:  df-bi 197  df-or 837  df-ex 1853
This theorem is referenced by:  eximp-surprise2  43055
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