Theorem sb5fALT

Description: Alternate version of sb5f . (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 4-Oct-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dfsb1.p5	$$⊢\theta \leftrightarrow \left(\left(x=y\to \phi \right)\wedge \exists x\left(x=y\wedge \phi \right)\right)$$
	sb6fALT.1	$$⊢Ⅎy\phi$$
Assertion	sb5fALT	$$⊢\theta \leftrightarrow \exists x\left(x=y\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dfsb1.p5	$$⊢\theta \leftrightarrow \left(\left(x=y\to \phi \right)\wedge \exists x\left(x=y\wedge \phi \right)\right)$$
2		sb6fALT.1	$$⊢Ⅎy\phi$$
3	1 2	sb6fALT	$$⊢\theta \leftrightarrow \forall x\left(x=y\to \phi \right)$$
4	2	equs45f	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)$$
5	3 4	bitr4i	$$⊢\theta \leftrightarrow \exists x\left(x=y\wedge \phi \right)$$