Theorem sb4aALT

Description: Alternate version of sb4a . (Contributed by NM, 2-Feb-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	dfsb1.p2	$$⊢\theta \leftrightarrow \left(\left(x=y\to \forall y\phi \right)\wedge \exists x\left(x=y\wedge \forall y\phi \right)\right)$$
Assertion	sb4aALT	$$⊢\theta \to \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dfsb1.p2	$$⊢\theta \leftrightarrow \left(\left(x=y\to \forall y\phi \right)\wedge \exists x\left(x=y\wedge \forall y\phi \right)\right)$$
2	1	sb1ALT	$$⊢\theta \to \exists x\left(x=y\wedge \forall y\phi \right)$$
3		equs5a	$$⊢\exists x\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$
4	2 3	syl	$$⊢\theta \to \forall x\left(x=y\to \phi \right)$$