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Metamath Proof Explorer

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) (Proof shortened by Wolf Lammen, 4-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		df-rex	$⊢ \exists x \in A \forall y \in B φ \leftrightarrow \exists x (x \in A \land \forall y \in B φ)$
2		nfv	$⊢ Ⅎ y x \in A$
3		nfra1	$⊢ Ⅎ y \forall y \in B φ$
4	2 3	nfan	$⊢ Ⅎ y (x \in A \land \forall y \in B φ)$
5	4	nfex	$⊢ Ⅎ y \exists x (x \in A \land \forall y \in B φ)$
6	1 5	nfxfr	$⊢ Ⅎ y \exists x \in A \forall y \in B φ$
7		rsp	$⊢ \forall y \in B φ \to (y \in B \to φ)$
8	7	com12	$⊢ y \in B \to (\forall y \in B φ \to φ)$
9	8	reximdv	$⊢ y \in B \to (\exists x \in A \forall y \in B φ \to \exists x \in A φ)$
10	9	com12	$⊢ \exists x \in A \forall y \in B φ \to (y \in B \to \exists x \in A φ)$
11	6 10	ralrimi	$⊢ \exists x \in A \forall y \in B φ \to \forall y \in B \exists x \in A φ$