Theorem sbftALT

Description: Alternate version of sbft . (Contributed by NM, 30-May-2009) (Revised by Mario Carneiro, 12-Oct-2016) (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	sbftALT	$$⊢Ⅎx\phi \to \left(\theta \leftrightarrow \phi \right)$$

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	spsbeALT	$$⊢\theta \to \exists x\phi$$
3		19.9t	$$⊢Ⅎx\phi \to \left(\exists x\phi \leftrightarrow \phi \right)$$
4	2 3	syl5ib	$$⊢Ⅎx\phi \to \left(\theta \to \phi \right)$$
5		nf5r	$$⊢Ⅎx\phi \to \left(\phi \to \forall x\phi \right)$$
6	1	stdpc4ALT	$$⊢\forall x\phi \to \theta$$
7	5 6	syl6	$$⊢Ⅎx\phi \to \left(\phi \to \theta \right)$$
8	4 7	impbid	$$⊢Ⅎx\phi \to \left(\theta \leftrightarrow \phi \right)$$