Theorem dvelimhw

Description: Proof of dvelimh without using ax-13 but with additional distinct variable conditions. (Contributed by NM, 1-Oct-2002) (Revised by Andrew Salmon, 21-Jul-2011) (Revised by NM, 1-Aug-2017) (Proof shortened by Wolf Lammen, 23-Dec-2018)

	Ref	Expression
Hypotheses	dvelimhw.1	$$⊢\phi \to \forall x\phi$$
	dvelimhw.2	$$⊢\psi \to \forall z\psi$$
	dvelimhw.3	$$⊢z=y\to \left(\phi \leftrightarrow \psi \right)$$
	dvelimhw.4	$$⊢\neg \forall xx=y\to \left(y=z\to \forall xy=z\right)$$
Assertion	dvelimhw	$$⊢\neg \forall xx=y\to \left(\psi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dvelimhw.1	$$⊢\phi \to \forall x\phi$$
2		dvelimhw.2	$$⊢\psi \to \forall z\psi$$
3		dvelimhw.3	$$⊢z=y\to \left(\phi \leftrightarrow \psi \right)$$
4		dvelimhw.4	$$⊢\neg \forall xx=y\to \left(y=z\to \forall xy=z\right)$$
5		nfv	$$⊢Ⅎz\neg \forall xx=y$$
6		equcom	$$⊢z=y\leftrightarrow y=z$$
7		nfna1	$$⊢Ⅎx\neg \forall xx=y$$
8	7 4	nf5d	$$⊢\neg \forall xx=y\to Ⅎxy=z$$
9	6 8	nfxfrd	$$⊢\neg \forall xx=y\to Ⅎxz=y$$
10	1	nf5i	$$⊢Ⅎx\phi$$
11	10	a1i	$$⊢\neg \forall xx=y\to Ⅎx\phi$$
12	9 11	nfimd	$$⊢\neg \forall xx=y\to Ⅎx\left(z=y\to \phi \right)$$
13	5 12	nfald	$$⊢\neg \forall xx=y\to Ⅎx\forall z\left(z=y\to \phi \right)$$
14	2 3	equsalhw	$$⊢\forall z\left(z=y\to \phi \right)\leftrightarrow \psi$$
15	14	nfbii	$$⊢Ⅎx\forall z\left(z=y\to \phi \right)\leftrightarrow Ⅎx\psi$$
16	13 15	sylib	$$⊢\neg \forall xx=y\to Ⅎx\psi$$
17	16	nf5rd	$$⊢\neg \forall xx=y\to \left(\psi \to \forall x\psi \right)$$