Theorem bj-eximcom

Description: A commuted form of exim which is sometimes posited as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-eximcom	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.27	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
2	1	aleximi	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 𝜓 ) )
3	2	com12	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )