Theorem sbequ2ALT

Description: Alternate version of sbequ2 . (Contributed by NM, 16-May-1993) (Proof shortened by Wolf Lammen, 25-Feb-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	sbequ2ALT	$$⊢x=y\to \left(\theta \to \phi \right)$$

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	simplbi	$$⊢\theta \to \left(x=y\to \phi \right)$$
3	2	com12	$$⊢x=y\to \left(\theta \to \phi \right)$$