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Metamath Proof Explorer

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 ) ) ) ) )