Theorem bj-cbveximt

Description: A lemma in closed form used to prove bj-cbvex in a weak axiomatization. (Contributed by BJ, 12-Mar-2023) Do not use 19.35 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-cbveximt	⊢ ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-exalim	⊢ ( ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∃ 𝑦 𝜒 → ( ∀ 𝑦 𝜑 → ∃ 𝑦 𝜓 ) ) )
2	1	alimi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ∀ 𝑥 ( ∃ 𝑦 𝜒 → ( ∀ 𝑦 𝜑 → ∃ 𝑦 𝜓 ) ) )
3		bj-alexim	⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜒 → ( ∀ 𝑦 𝜑 → ∃ 𝑦 𝜓 ) ) → ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ∃ 𝑥 ∀ 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 ) ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ∃ 𝑥 ∀ 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 ) ) )
5		exim	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ∀ 𝑦 𝜑 ) )
6		imim2	⊢ ( ( ∃ 𝑥 ∀ 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 ) → ( ( ∃ 𝑥 𝜑 → ∃ 𝑥 ∀ 𝑦 𝜑 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 ) ) )
7		imim1	⊢ ( ( ∃ 𝑥 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 ) → ( ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) )
8	5 6 7	syl56	⊢ ( ( ∃ 𝑥 ∀ 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 ) → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) ) )
9	4 8	syl6com	⊢ ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) ) ) )