Theorem ex-natded9.26-2

Description: A more efficient proof of Theorem 9.26 of Clemente p. 45. Compare with ex-natded9.26 . (Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ex-natded9.26.1	$$⊢\phi \to \exists x\forall y\psi$$
Assertion	ex-natded9.26-2	$$⊢\phi \to \forall y\exists x\psi$$

Proof

Step	Hyp	Ref	Expression
1		ex-natded9.26.1	$$⊢\phi \to \exists x\forall y\psi$$
2		sp	$$⊢\forall y\psi \to \psi$$
3	2	eximi	$$⊢\exists x\forall y\psi \to \exists x\psi$$
4	1 3	syl	$$⊢\phi \to \exists x\psi$$
5	4	alrimiv	$$⊢\phi \to \forall y\exists x\psi$$