Theorem sb4ALT

Description: Alternate version of one implication of sb4b . (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	sb4ALT	$$⊢\neg \forall xx=y\to \left(\theta \to \forall x\left(x=y\to \phi \right)\right)$$

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	sb1ALT	$$⊢\theta \to \exists x\left(x=y\wedge \phi \right)$$
3		equs5	$$⊢\neg \forall xx=y\to \left(\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$
4	2 3	syl5ib	$$⊢\neg \forall xx=y\to \left(\theta \to \forall x\left(x=y\to \phi \right)\right)$$