Theorem spsbce-2

Description: Theorem *11.36 in WhiteheadRussell p. 162. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	spsbce-2	$$⊢\left[z/x\right]\left[w/y\right]\phi \to \exists x\exists y\phi$$

Proof

Step	Hyp	Ref	Expression
1		spsbe	$$⊢\left[z/x\right]\left[w/y\right]\phi \to \exists x\left[w/y\right]\phi$$
2		spsbe	$$⊢\left[w/y\right]\phi \to \exists y\phi$$
3	2	eximi	$$⊢\exists x\left[w/y\right]\phi \to \exists x\exists y\phi$$
4	1 3	syl	$$⊢\left[z/x\right]\left[w/y\right]\phi \to \exists x\exists y\phi$$