Theorem exellim

Description: Closed form of exellimddv . See also exlimim for a more general theorem. (Contributed by ML, 17-Jul-2020)

	Ref	Expression
Assertion	exellim	⊢ ( ( ∃ 𝑥 𝑥 ∈ 𝐴 ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → 𝜑 )

Proof

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