Theorem bj-aleximiALT

Description: Alternate proof of aleximi from exim , which is sometimes used as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bj-aleximiALT.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion	bj-aleximiALT	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-aleximiALT.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2	1	alimi	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) )
3		bj-eximALT	⊢ ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )