Theorem eximp-surprise 44250

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 839, 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 44251. See also alimp-surprise 44246 and empty-surprise 44248. (Contributed by David A. Wheeler, 17-Oct-2018.)

Assertion

Ref	Expression
eximp-surprise	⊢ (∃𝑥(𝜑 → 𝜓) ↔ ∃𝑥(¬ 𝜑 ∨ 𝜓))

Proof of Theorem eximp-surprise

Step	Hyp	Ref	Expression
1		imor 839	. 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
2	1	exbii 1810	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ ∃𝑥(¬ 𝜑 ∨ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∨ wo 833  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772

This theorem depends on definitions:  df-bi 199  df-or 834  df-ex 1743

This theorem is referenced by:  eximp-surprise2  44251