Theorem nfsb4ALT

Description: Alternate version of nfsb4 . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 4-Oct-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dfsb1.p5	$$⊢\theta \leftrightarrow \left(\left(x=y\to \phi \right)\wedge \exists x\left(x=y\wedge \phi \right)\right)$$
	nfsb4ALT.1	$$⊢Ⅎz\phi$$
Assertion	nfsb4ALT	$$⊢\neg \forall zz=y\to Ⅎz\theta$$

Proof

Step	Hyp	Ref	Expression
1		dfsb1.p5	$$⊢\theta \leftrightarrow \left(\left(x=y\to \phi \right)\wedge \exists x\left(x=y\wedge \phi \right)\right)$$
2		nfsb4ALT.1	$$⊢Ⅎz\phi$$
3	1	nfsb4tALT	$$⊢\forall xℲz\phi \to \left(\neg \forall zz=y\to Ⅎz\theta \right)$$
4	3 2	mpg	$$⊢\neg \forall zz=y\to Ⅎz\theta$$