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Theorem axc5c4c711toc7 37430
Description: Rederivation of axc7 2115 from axc5c4c711 37427. Note that neither axc7 2115 nor ax-11 2020 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c4c711toc7 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Proof of Theorem axc5c4c711toc7
StepHypRef Expression
1 ax-1 6 . . . . . . . 8 (𝜑 → (∀𝑥(𝜑𝜑) → 𝜑))
21alimi 1729 . . . . . . 7 (∀𝑥𝜑 → ∀𝑥(∀𝑥(𝜑𝜑) → 𝜑))
32axc4i 2114 . . . . . 6 (∀𝑥𝜑 → ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑))
43con3i 148 . . . . 5 (¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ¬ ∀𝑥𝜑)
54alimi 1729 . . . 4 (∀𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ∀𝑥 ¬ ∀𝑥𝜑)
65sps 2041 . . 3 (∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ∀𝑥 ¬ ∀𝑥𝜑)
76con3i 148 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑))
8 pm2.21 118 . . . 4 (¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → (∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ((𝜑𝜑) → ∀𝑥(∀𝑥(𝜑𝜑) → 𝜑))))
9 axc5c4c711 37427 . . . 4 ((∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ((𝜑𝜑) → ∀𝑥(∀𝑥(𝜑𝜑) → 𝜑))) → (∀𝑥(𝜑𝜑) → ∀𝑥𝜑))
108, 9syl 17 . . 3 (¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → (∀𝑥(𝜑𝜑) → ∀𝑥𝜑))
11 sp 2039 . . 3 (∀𝑥𝜑𝜑)
1210, 11syl6 34 . 2 (¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → (∀𝑥(𝜑𝜑) → 𝜑))
13 pm2.27 40 . . 3 (∀𝑥(𝜑𝜑) → ((∀𝑥(𝜑𝜑) → 𝜑) → 𝜑))
14 id 22 . . 3 (𝜑𝜑)
1513, 14mpg 1714 . 2 ((∀𝑥(𝜑𝜑) → 𝜑) → 𝜑)
167, 12, 153syl 18 1 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
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-10 2005  ax-11 2020  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695  df-nf 1700
This theorem is referenced by:  axc5c4c711to11  37431
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