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Theorem axc5c4c711to11 39132
Description: Rederivation of ax-11 2190 from axc5c4c711 39128. Note that ax-11 2190 is not required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c4c711to11 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Proof of Theorem axc5c4c711to11
StepHypRef Expression
1 ax-1 6 . . 3 (𝜑 → (∀𝑦(𝜑𝜑) → 𝜑))
212alimi 1888 . 2 (∀𝑥𝑦𝜑 → ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑))
3 axc5c4c711toc7 39131 . . . 4 (¬ ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑))
43con4i 114 . . 3 (∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑))
5 pm2.21 121 . . . . . . 7 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → (∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ((𝜑𝜑) → ∀𝑦(∀𝑦(𝜑𝜑) → 𝜑))))
6 axc5c4c711 39128 . . . . . . . 8 ((∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ((𝜑𝜑) → ∀𝑦(∀𝑦(𝜑𝜑) → 𝜑))) → (∀𝑦(𝜑𝜑) → ∀𝑦𝜑))
7 sp 2207 . . . . . . . 8 (∀𝑦𝜑𝜑)
86, 7syl6 35 . . . . . . 7 ((∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ((𝜑𝜑) → ∀𝑦(∀𝑦(𝜑𝜑) → 𝜑))) → (∀𝑦(𝜑𝜑) → 𝜑))
95, 8syl 17 . . . . . 6 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → (∀𝑦(𝜑𝜑) → 𝜑))
109alimi 1887 . . . . 5 (∀𝑥 ¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ∀𝑥(∀𝑦(𝜑𝜑) → 𝜑))
11 axc5c4c711toc7 39131 . . . . 5 (¬ ∀𝑥 ¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ∀𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑))
1210, 11nsyl4 157 . . . 4 (¬ ∀𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ∀𝑥(∀𝑦(𝜑𝜑) → 𝜑))
1312alimi 1887 . . 3 (∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ∀𝑦𝑥(∀𝑦(𝜑𝜑) → 𝜑))
144, 13syl 17 . 2 (∀𝑥𝑦(∀𝑦(𝜑𝜑) → 𝜑) → ∀𝑦𝑥(∀𝑦(𝜑𝜑) → 𝜑))
15 pm2.27 42 . . . 4 (∀𝑦(𝜑𝜑) → ((∀𝑦(𝜑𝜑) → 𝜑) → 𝜑))
16 id 22 . . . 4 (𝜑𝜑)
1715, 16mpg 1872 . . 3 ((∀𝑦(𝜑𝜑) → 𝜑) → 𝜑)
18172alimi 1888 . 2 (∀𝑦𝑥(∀𝑦(𝜑𝜑) → 𝜑) → ∀𝑦𝑥𝜑)
192, 14, 183syl 18 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-or 835  df-ex 1853  df-nf 1858
This theorem is referenced by: (None)
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