Theorem sb6ALT

Description: Alternate version of sb6 . (Contributed by NM, 18-Aug-1993) (Proof shortened by Wolf Lammen, 21-Sep-2018) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	dfsb1.ph	$$⊢\theta \leftrightarrow \left(\left(x=y\to \phi \right)\wedge \exists x\left(x=y\wedge \phi \right)\right)$$
Assertion	sb6ALT	$$⊢\theta \leftrightarrow \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dfsb1.ph	$$⊢\theta \leftrightarrow \left(\left(x=y\to \phi \right)\wedge \exists x\left(x=y\wedge \phi \right)\right)$$
2	1	sb4vOLDALT	$$⊢\theta \to \forall x\left(x=y\to \phi \right)$$
3	1	sb2vOLDALT	$$⊢\forall x\left(x=y\to \phi \right)\to \theta$$
4	2 3	impbii	$$⊢\theta \leftrightarrow \forall x\left(x=y\to \phi \right)$$