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Metamath Proof Explorer

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	⊢ ( 𝜑 → ∃ 𝑥 ∀ 𝑦 𝜓 )
	Assertion	ex-natded9.26-2	⊢ ( 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ex-natded9.26.1	⊢ ( 𝜑 → ∃ 𝑥 ∀ 𝑦 𝜓 )
2		sp	⊢ ( ∀ 𝑦 𝜓 → 𝜓 )
3	2	eximi	⊢ ( ∃ 𝑥 ∀ 𝑦 𝜓 → ∃ 𝑥 𝜓 )
4	1 3	syl	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
5	4	alrimiv	⊢ ( 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜓 )