Theorem ex-natded9.26-2 30440

Description: A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 30439. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ex-natded9.26.1	⊢ (𝜑 → ∃𝑥∀𝑦𝜓)

Assertion

Ref	Expression
ex-natded9.26-2	⊢ (𝜑 → ∀𝑦∃𝑥𝜓)

Distinct variable group:   𝑥,𝑦,𝜑

Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem ex-natded9.26-2

Step	Hyp	Ref	Expression
1		ex-natded9.26.1	. . 3 ⊢ (𝜑 → ∃𝑥∀𝑦𝜓)
2		sp 2182	. . . 4 ⊢ (∀𝑦𝜓 → 𝜓)
3	2	eximi 1834	. . 3 ⊢ (∃𝑥∀𝑦𝜓 → ∃𝑥𝜓)
4	1, 3	syl 17	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
5	4	alrimiv 1926	1 ⊢ (𝜑 → ∀𝑦∃𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-ex 1779

This theorem is referenced by: (None)