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Theorem axc5c4c711toc7 40736
Description: Rederivation of axc7 2332 from axc5c4c711 40733. Note that neither axc7 2332 nor ax-11 2157 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 1808 . . . . . . 7 (∀𝑥𝜑 → ∀𝑥(∀𝑥(𝜑𝜑) → 𝜑))
32axc4i 2337 . . . . . 6 (∀𝑥𝜑 → ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑))
43con3i 157 . . . . 5 (¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ¬ ∀𝑥𝜑)
54alimi 1808 . . . 4 (∀𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ∀𝑥 ¬ ∀𝑥𝜑)
65sps 2180 . . 3 (∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ∀𝑥 ¬ ∀𝑥𝜑)
76con3i 157 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑))
8 pm2.21 123 . . . 4 (¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → (∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → ((𝜑𝜑) → ∀𝑥(∀𝑥(𝜑𝜑) → 𝜑))))
9 axc5c4c711 40733 . . . 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 2178 . . 3 (∀𝑥𝜑𝜑)
1210, 11syl6 35 . 2 (¬ ∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥(𝜑𝜑) → 𝜑) → (∀𝑥(𝜑𝜑) → 𝜑))
13 pm2.27 42 . . 3 (∀𝑥(𝜑𝜑) → ((∀𝑥(𝜑𝜑) → 𝜑) → 𝜑))
14 id 22 . . 3 (𝜑𝜑)
1513, 14mpg 1794 . 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 1531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777  df-nf 1781
This theorem is referenced by:  axc5c4c711to11  40737
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