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Theorem ex-natded9.26-2 26463
Description: A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 26462. (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 2040 . . . 4 (∀𝑦𝜓𝜓)
32eximi 1751 . . 3 (∃𝑥𝑦𝜓 → ∃𝑥𝜓)
41, 3syl 17 . 2 (𝜑 → ∃𝑥𝜓)
54alrimiv 1841 1 (𝜑 → ∀𝑦𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by: (None)
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