Theorem nfrexdg

Description: Deduction version of nfrexg . Usage of this theorem is discouraged because it depends on ax-13 . See nfrexd for a version with a disjoint variable condition, but not requiring ax-13 . (Contributed by Mario Carneiro, 14-Oct-2016) (New usage is discouraged.)

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

Proof

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