Theorem spsbeALT

Description: Alternate version of spsbe . (Contributed by NM, 29-Jun-1993) (Proof shortened by Wolf Lammen, 3-May-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	spsbeALT	$$⊢\theta \to \exists x\phi$$

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	sb1ALT	$$⊢\theta \to \exists x\left(x=y\wedge \phi \right)$$
3		exsimpr	$$⊢\exists x\left(x=y\wedge \phi \right)\to \exists x\phi$$
4	2 3	syl	$$⊢\theta \to \exists x\phi$$