Theorem bj-cbveximt 34800

Description: A lemma in closed form used to prove bj-cbvex 34810 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1883 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cbveximt	⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))

Proof of Theorem bj-cbveximt

Step	Hyp	Ref	Expression
1		bj-exalim 34792	. . . 4 ⊢ (∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑦𝜒 → (∀𝑦𝜑 → ∃𝑦𝜓)))
2	1	alimi 1817	. . 3 ⊢ (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → ∀𝑥(∃𝑦𝜒 → (∀𝑦𝜑 → ∃𝑦𝜓)))
3		bj-alexim 34787	. . 3 ⊢ (∀𝑥(∃𝑦𝜒 → (∀𝑦𝜑 → ∃𝑦𝜓)) → (∀𝑥∃𝑦𝜒 → (∃𝑥∀𝑦𝜑 → ∃𝑥∃𝑦𝜓)))
4	2, 3	syl 17	. 2 ⊢ (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥∃𝑦𝜒 → (∃𝑥∀𝑦𝜑 → ∃𝑥∃𝑦𝜓)))
5		exim 1839	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑦𝜑))
6		imim2 58	. . 3 ⊢ ((∃𝑥∀𝑦𝜑 → ∃𝑥∃𝑦𝜓) → ((∃𝑥𝜑 → ∃𝑥∀𝑦𝜑) → (∃𝑥𝜑 → ∃𝑥∃𝑦𝜓)))
7		imim1 83	. . 3 ⊢ ((∃𝑥𝜑 → ∃𝑥∃𝑦𝜓) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))
8	5, 6, 7	syl56 36	. 2 ⊢ ((∃𝑥∀𝑦𝜑 → ∃𝑥∃𝑦𝜓) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓))))
9	4, 8	syl6com 37	1 ⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815

This theorem depends on definitions:  df-bi 206  df-ex 1786

This theorem is referenced by:  bj-cbvexim  34806