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Theorem axc5c4c711 39098
Description: Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1880 as the inference rule. This proof extends the idea of axc5c711 34694 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 2308 . . 3 (∀𝑦(∀𝑦𝜑𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
2 hbn1 2188 . . . . 5 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑦 ¬ ∀𝑦(∀𝑦𝜑𝜓))
3 axc7 2310 . . . . . 6 (¬ ∀𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → ∀𝑦(∀𝑦𝜑𝜓))
43con1i 146 . . . . 5 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
52, 4alrimih 1911 . . . 4 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑦𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
6 ax-11 2203 . . . 4 (∀𝑦𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
75, 6syl 17 . . 3 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
81, 7nsyl4 157 . 2 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
9 pm2.21 121 . . . 4 𝜑 → (𝜑 → ∀𝑦𝜓))
109spsd 2224 . . 3 𝜑 → (∀𝑦𝜑 → ∀𝑦𝜓))
1110, 1ja 174 . 2 ((𝜑 → ∀𝑦(∀𝑦𝜑𝜓)) → (∀𝑦𝜑 → ∀𝑦𝜓))
128, 11ja 174 1 ((∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-10 2187  ax-11 2203  ax-12 2216
This theorem depends on definitions:  df-bi 198  df-ex 1860
This theorem is referenced by:  axc5c4c711toc5  39099  axc5c4c711toc4  39100  axc5c4c711toc7  39101  axc5c4c711to11  39102
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