Theorem r19.12

Description: Restricted quantifier version of 19.12 . (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) Avoid ax-13 , ax-ext . (Revised by Wolf Lammen, 17-Jun-2023)

	Ref	Expression
Assertion	r19.12	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐵 𝜑 ) )
2		nfv	⊢ Ⅎ 𝑦 𝑥 ∈ 𝐴
3		nfra1	⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝐵 𝜑
4	2 3	nfan	⊢ Ⅎ 𝑦 ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐵 𝜑 )
5	4	nfex	⊢ Ⅎ 𝑦 ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐵 𝜑 )
6	1 5	nfxfr	⊢ Ⅎ 𝑦 ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑
7		ax-1	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ( 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ) )
8		rsp	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜑 → ( 𝑦 ∈ 𝐵 → 𝜑 ) )
9	8	com12	⊢ ( 𝑦 ∈ 𝐵 → ( ∀ 𝑦 ∈ 𝐵 𝜑 → 𝜑 ) )
10	9	reximdv	⊢ ( 𝑦 ∈ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜑 ) )
11	7 10	sylcom	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ( 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 𝜑 ) )
12	6 11	ralrimi	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝜑 )