Theorem r19.12OLD

Description: Obsolete version of r19.12 as of 19-May-2023. (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	r19.12OLD	$$⊢\exists x\in A\forall y\in B\phi \to \forall y\in B\exists x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
2		nfra1	$$⊢Ⅎy\forall y\in B\phi$$
3	1 2	nfrex	$$⊢Ⅎy\exists x\in A\forall y\in B\phi$$
4		ax-1	$$⊢\exists x\in A\forall y\in B\phi \to \left(y\in B\to \exists x\in A\forall y\in B\phi \right)$$
5	3 4	ralrimi	$$⊢\exists x\in A\forall y\in B\phi \to \forall y\in B\exists x\in A\forall y\in B\phi$$
6		rsp	$$⊢\forall y\in B\phi \to \left(y\in B\to \phi \right)$$
7	6	com12	$$⊢y\in B\to \left(\forall y\in B\phi \to \phi \right)$$
8	7	reximdv	$$⊢y\in B\to \left(\exists x\in A\forall y\in B\phi \to \exists x\in A\phi \right)$$
9	8	ralimia	$$⊢\forall y\in B\exists x\in A\forall y\in B\phi \to \forall y\in B\exists x\in A\phi$$
10	5 9	syl	$$⊢\exists x\in A\forall y\in B\phi \to \forall y\in B\exists x\in A\phi$$