Theorem bj-cbvalimt

Description: A lemma in closed form used to prove bj-cbval 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-cbvalimt	⊢ ( ∀ 𝑦 ∃ 𝑥 𝜒 → ( ∀ 𝑦 ∀ 𝑥 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) → ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		exim	⊢ ( ∀ 𝑥 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∃ 𝑥 𝜒 → ∃ 𝑥 ( 𝜑 → 𝜓 ) ) )
2	1	al2imi	⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑦 ∃ 𝑥 𝜒 → ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) ) )
3		pm2.27	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
4	3	aleximi	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 𝜓 ) )
5	4	com12	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
6	5	alimi	⊢ ( ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) → ∀ 𝑦 ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
7		alim	⊢ ( ∀ 𝑦 ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) → ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜓 ) )
8		alim	⊢ ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∀ 𝑦 ∃ 𝑥 𝜓 → ∀ 𝑦 𝜓 ) )
9		imim1	⊢ ( ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜓 ) → ( ( ∀ 𝑦 ∃ 𝑥 𝜓 → ∀ 𝑦 𝜓 ) → ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
10		imim2	⊢ ( ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) → ( ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
11	8 9 10	syl56	⊢ ( ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜓 ) → ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) ) )
12	11	com23	⊢ ( ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜓 ) → ( ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) → ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) ) )
13	6 7 12	3syl	⊢ ( ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) → ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) ) )
14	2 13	syl6com	⊢ ( ∀ 𝑦 ∃ 𝑥 𝜒 → ( ∀ 𝑦 ∀ 𝑥 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) → ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) ) ) )