Theorem bj-nnfext

Description: See nfex and bj-nfext . (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-nnfext	$$⊢\forall xℲ\text{'}y\phi \to Ⅎ\text{'}y\exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-bj-nnf	$$⊢Ⅎ\text{'}y\phi \leftrightarrow \left(\left(\exists y\phi \to \phi \right)\wedge \left(\phi \to \forall y\phi \right)\right)$$
2	1	albii	$$⊢\forall xℲ\text{'}y\phi \leftrightarrow \forall x\left(\left(\exists y\phi \to \phi \right)\wedge \left(\phi \to \forall y\phi \right)\right)$$
3		simpl	$$⊢\left(\left(\exists y\phi \to \phi \right)\wedge \left(\phi \to \forall y\phi \right)\right)\to \left(\exists y\phi \to \phi \right)$$
4	3	alimi	$$⊢\forall x\left(\left(\exists y\phi \to \phi \right)\wedge \left(\phi \to \forall y\phi \right)\right)\to \forall x\left(\exists y\phi \to \phi \right)$$
5		bj-nnflemee	$$⊢\forall x\left(\exists y\phi \to \phi \right)\to \left(\exists y\exists x\phi \to \exists x\phi \right)$$
6	4 5	syl	$$⊢\forall x\left(\left(\exists y\phi \to \phi \right)\wedge \left(\phi \to \forall y\phi \right)\right)\to \left(\exists y\exists x\phi \to \exists x\phi \right)$$
7	2 6	sylbi	$$⊢\forall xℲ\text{'}y\phi \to \left(\exists y\exists x\phi \to \exists x\phi \right)$$
8		simpr	$$⊢\left(\left(\exists y\phi \to \phi \right)\wedge \left(\phi \to \forall y\phi \right)\right)\to \left(\phi \to \forall y\phi \right)$$
9	8	alimi	$$⊢\forall x\left(\left(\exists y\phi \to \phi \right)\wedge \left(\phi \to \forall y\phi \right)\right)\to \forall x\left(\phi \to \forall y\phi \right)$$
10		bj-nnflemae	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \left(\exists x\phi \to \forall y\exists x\phi \right)$$
11	9 10	syl	$$⊢\forall x\left(\left(\exists y\phi \to \phi \right)\wedge \left(\phi \to \forall y\phi \right)\right)\to \left(\exists x\phi \to \forall y\exists x\phi \right)$$
12	2 11	sylbi	$$⊢\forall xℲ\text{'}y\phi \to \left(\exists x\phi \to \forall y\exists x\phi \right)$$
13		df-bj-nnf	$$⊢Ⅎ\text{'}y\exists x\phi \leftrightarrow \left(\left(\exists y\exists x\phi \to \exists x\phi \right)\wedge \left(\exists x\phi \to \forall y\exists x\phi \right)\right)$$
14	7 12 13	sylanbrc	$$⊢\forall xℲ\text{'}y\phi \to Ⅎ\text{'}y\exists x\phi$$