Theorem bj-cbvexim

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

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

Proof

Step	Hyp	Ref	Expression
1		ax5e	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 )
2		ax-5	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3	2	ax-gen	⊢ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 )
4		bj-cbveximt	⊢ ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) ) ) )
5	4	com3l	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) ) ) )
6	5	com14	⊢ ( ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 ) → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) ) ) )
7	1 3 6	mp2	⊢ ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) )