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Theorem axc5c4c711 38919
 Description: Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1762 as the inference rule. This proof extends the idea of axc5c711 34522 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
Assertion
Ref Expression
axc5c4c711 ((∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))

Proof of Theorem axc5c4c711
StepHypRef Expression
1 axc4 2168 . . 3 (∀𝑦(∀𝑦𝜑𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
2 hbn1 2060 . . . . 5 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑦 ¬ ∀𝑦(∀𝑦𝜑𝜓))
3 axc7 2170 . . . . . 6 (¬ ∀𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → ∀𝑦(∀𝑦𝜑𝜓))
43con1i 144 . . . . 5 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
52, 4alrimih 1791 . . . 4 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑦𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
6 ax-11 2074 . . . 4 (∀𝑦𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
75, 6syl 17 . . 3 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
81, 7nsyl4 156 . 2 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
9 pm2.21 120 . . . 4 𝜑 → (𝜑 → ∀𝑦𝜓))
109spsd 2095 . . 3 𝜑 → (∀𝑦𝜑 → ∀𝑦𝜓))
1110, 1ja 173 . 2 ((𝜑 → ∀𝑦(∀𝑦𝜑𝜓)) → (∀𝑦𝜑 → ∀𝑦𝜓))
128, 11ja 173 1 ((∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-ex 1745 This theorem is referenced by:  axc5c4c711toc5  38920  axc5c4c711toc4  38921  axc5c4c711toc7  38922  axc5c4c711to11  38923
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