Theorem bj-cbvalim

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

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

Proof

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