Theorem dvelimh

Description: Version of dvelim without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 . Check out dvelimhw for a version requiring fewer axioms. (Contributed by NM, 1-Oct-2002) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvelimh.1	$$⊢\phi \to \forall x\phi$$
	dvelimh.2	$$⊢\psi \to \forall z\psi$$
	dvelimh.3	$$⊢z=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	dvelimh	$$⊢\neg \forall xx=y\to \left(\psi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dvelimh.1	$$⊢\phi \to \forall x\phi$$
2		dvelimh.2	$$⊢\psi \to \forall z\psi$$
3		dvelimh.3	$$⊢z=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1	nf5i	$$⊢Ⅎx\phi$$
5	2	nf5i	$$⊢Ⅎz\psi$$
6	4 5 3	dvelimf	$$⊢\neg \forall xx=y\to Ⅎx\psi$$
7	6	nf5rd	$$⊢\neg \forall xx=y\to \left(\psi \to \forall x\psi \right)$$