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	$$⊢\exists x\in A\forall y\in B\phi \to \forall y\in B\exists x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-rex	$$⊢\exists x\in A\forall y\in B\phi \leftrightarrow \exists x\left(x\in A\wedge \forall y\in B\phi \right)$$
2		nfv	$$⊢Ⅎyx\in A$$
3		nfra1	$$⊢Ⅎy\forall y\in B\phi$$
4	2 3	nfan	$$⊢Ⅎy\left(x\in A\wedge \forall y\in B\phi \right)$$
5	4	nfex	$$⊢Ⅎy\exists x\left(x\in A\wedge \forall y\in B\phi \right)$$
6	1 5	nfxfr	$$⊢Ⅎy\exists x\in A\forall y\in B\phi$$
7		ax-1	$$⊢\exists x\in A\forall y\in B\phi \to \left(y\in B\to \exists x\in A\forall y\in B\phi \right)$$
8		rsp	$$⊢\forall y\in B\phi \to \left(y\in B\to \phi \right)$$
9	8	com12	$$⊢y\in B\to \left(\forall y\in B\phi \to \phi \right)$$
10	9	reximdv	$$⊢y\in B\to \left(\exists x\in A\forall y\in B\phi \to \exists x\in A\phi \right)$$
11	7 10	sylcom	$$⊢\exists x\in A\forall y\in B\phi \to \left(y\in B\to \exists x\in A\phi \right)$$
12	6 11	ralrimi	$$⊢\exists x\in A\forall y\in B\phi \to \forall y\in B\exists x\in A\phi$$