Theorem bj-cbveximi

Description: An equality-free general instance of one half of a precise form of bj-cbvex . (Contributed by BJ, 12-Mar-2023) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbvalimi.maj	⊢ ( 𝜒 → ( 𝜑 → 𝜓 ) )
	bj-cbveximi.denote	⊢ ∀ 𝑥 ∃ 𝑦 𝜒
Assertion	bj-cbveximi	⊢ ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-cbvalimi.maj	⊢ ( 𝜒 → ( 𝜑 → 𝜓 ) )
2		bj-cbveximi.denote	⊢ ∀ 𝑥 ∃ 𝑦 𝜒
3	1	gen2	⊢ ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) )
4		bj-cbvexim	⊢ ( ∀ 𝑥 ∃ 𝑦 𝜒 → ( ∀ 𝑥 ∀ 𝑦 ( 𝜒 → ( 𝜑 → 𝜓 ) ) → ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 ) ) )
5	2 3 4	mp2	⊢ ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 )