Theorem bj-cbvexim 34827

Description: A lemma used to prove bj-cbvex 34831 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-cbvexim

Step	Hyp	Ref	Expression
1		ax5e 1915	. 2 ⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)
2		ax-5 1913	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	ax-gen 1798	. 2 ⊢ ∀𝑥(𝜑 → ∀𝑦𝜑)
4		bj-cbveximt 34821	. . . 4 ⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
5	4	com3l 89	. . 3 ⊢ (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥∃𝑦𝜒 → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
6	5	com14 96	. 2 ⊢ ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓)))))
7	1, 3, 6	mp2 9	1 ⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓)))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  bj-cbveximi  34829