Theorem sbequ12ALT

Description: Alternate version of sbequ12 . (Contributed by NM, 14-May-1993) (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	sbequ12ALT	$$⊢x=y\to \left(\phi \leftrightarrow \theta \right)$$

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	sbequ1ALT	$$⊢x=y\to \left(\phi \to \theta \right)$$
3	1	sbequ2ALT	$$⊢x=y\to \left(\theta \to \phi \right)$$
4	2 3	impbid	$$⊢x=y\to \left(\phi \leftrightarrow \theta \right)$$