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Theorem ex-natded9.26-2 26669
 Description: A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 26668. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ex-natded9.26.1 (𝜑 → ∃𝑥𝑦𝜓)
Assertion
Ref Expression
ex-natded9.26-2 (𝜑 → ∀𝑦𝑥𝜓)
Distinct variable group:   𝑥,𝑦,𝜑
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem ex-natded9.26-2
StepHypRef Expression
1 ex-natded9.26.1 . . 3 (𝜑 → ∃𝑥𝑦𝜓)
2 sp 2041 . . . 4 (∀𝑦𝜓𝜓)
32eximi 1752 . . 3 (∃𝑥𝑦𝜓 → ∃𝑥𝜓)
41, 3syl 17 . 2 (𝜑 → ∃𝑥𝜓)
54alrimiv 1842 1 (𝜑 → ∀𝑦𝑥𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by: (None)
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