Theorem eximal

Description: An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. An interesting case is when the same formula is substituted for both ph and ps , since then both implications express a type of nonfreeness. See also alimex . (Contributed by BJ, 12-May-2019)

	Ref	Expression
Assertion	eximal	⊢ ( ( ∃ 𝑥 𝜑 → 𝜓 ) ↔ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ex	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
2	1	imbi1i	⊢ ( ( ∃ 𝑥 𝜑 → 𝜓 ) ↔ ( ¬ ∀ 𝑥 ¬ 𝜑 → 𝜓 ) )
3		con1b	⊢ ( ( ¬ ∀ 𝑥 ¬ 𝜑 → 𝜓 ) ↔ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜑 ) )
4	2 3	bitri	⊢ ( ( ∃ 𝑥 𝜑 → 𝜓 ) ↔ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜑 ) )