Theorem eximp-surprise

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 , such an expression can be rewritten usingnot withor - 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 . See also alimp-surprise and empty-surprise . (Contributed by David A. Wheeler, 17-Oct-2018)

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

Proof

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