Theorem dvelimdf

Description: Deduction form of dvelimf . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 7-Apr-2004) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvelimdf.1	$$⊢Ⅎx\phi$$
	dvelimdf.2	$$⊢Ⅎz\phi$$
	dvelimdf.3	$$⊢\phi \to Ⅎx\psi$$
	dvelimdf.4	$$⊢\phi \to Ⅎz\chi$$
	dvelimdf.5	$$⊢\phi \to \left(z=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	dvelimdf	$$⊢\phi \to \left(\neg \forall xx=y\to Ⅎx\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dvelimdf.1	$$⊢Ⅎx\phi$$
2		dvelimdf.2	$$⊢Ⅎz\phi$$
3		dvelimdf.3	$$⊢\phi \to Ⅎx\psi$$
4		dvelimdf.4	$$⊢\phi \to Ⅎz\chi$$
5		dvelimdf.5	$$⊢\phi \to \left(z=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
6	1 3	nfim1	$$⊢Ⅎx\left(\phi \to \psi \right)$$
7	2 4	nfim1	$$⊢Ⅎz\left(\phi \to \chi \right)$$
8	5	com12	$$⊢z=y\to \left(\phi \to \left(\psi \leftrightarrow \chi \right)\right)$$
9	8	pm5.74d	$$⊢z=y\to \left(\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)\right)$$
10	6 7 9	dvelimf	$$⊢\neg \forall xx=y\to Ⅎx\left(\phi \to \chi \right)$$
11		pm5.5	$$⊢\phi \to \left(\left(\phi \to \chi \right)\leftrightarrow \chi \right)$$
12	1 11	nfbidf	$$⊢\phi \to \left(Ⅎx\left(\phi \to \chi \right)\leftrightarrow Ⅎx\chi \right)$$
13	10 12	syl5ib	$$⊢\phi \to \left(\neg \forall xx=y\to Ⅎx\chi \right)$$