Theorem exlimim

Description: Closed form of exlimimd . (Contributed by ML, 17-Jul-2020)

	Ref	Expression
Assertion	exlimim	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝜑 → 𝜓 )
2		nfv	⊢ Ⅎ 𝑥 𝜓
3		sp	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) )
4	1 2 3	exlimd	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → 𝜓 ) )
5	4	impcom	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) → 𝜓 )