Theorem nfrexdw

Description: Deduction version of nfrexw . (Contributed by Mario Carneiro, 14-Oct-2016) Add disjoint variable condition to avoid ax-13 . See nfrexd for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypotheses	nfraldw.1	$$⊢Ⅎy\phi$$
	nfraldw.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
	nfraldw.3	$$⊢\phi \to Ⅎx\psi$$
Assertion	nfrexdw	$$⊢\phi \to Ⅎx\exists y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		nfraldw.1	$$⊢Ⅎy\phi$$
2		nfraldw.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
3		nfraldw.3	$$⊢\phi \to Ⅎx\psi$$
4		dfrex2	$$⊢\exists y\in A\psi \leftrightarrow \neg \forall y\in A\neg \psi$$
5	3	nfnd	$$⊢\phi \to Ⅎx\neg \psi$$
6	1 2 5	nfraldw	$$⊢\phi \to Ⅎx\forall y\in A\neg \psi$$
7	6	nfnd	$$⊢\phi \to Ⅎx\neg \forall y\in A\neg \psi$$
8	4 7	nfxfrd	$$⊢\phi \to Ⅎx\exists y\in A\psi$$