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Theorem axc5c4c711 40094
Description: Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1758 as the inference rule. This proof extends the idea of axc5c711 35447 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 2259 . . 3 (∀𝑦(∀𝑦𝜑𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
2 hbn1 2078 . . . . 5 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑦 ¬ ∀𝑦(∀𝑦𝜑𝜓))
3 axc7 2255 . . . . . 6 (¬ ∀𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → ∀𝑦(∀𝑦𝜑𝜓))
43con1i 147 . . . . 5 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
52, 4alrimih 1786 . . . 4 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑦𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
6 ax-11 2091 . . . 4 (∀𝑦𝑥 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
75, 6syl 17 . . 3 (¬ ∀𝑦(∀𝑦𝜑𝜓) → ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓))
81, 7nsyl4 158 . 2 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
9 pm2.21 121 . . . 4 𝜑 → (𝜑 → ∀𝑦𝜓))
109spsd 2113 . . 3 𝜑 → (∀𝑦𝜑 → ∀𝑦𝜓))
1110, 1ja 175 . 2 ((𝜑 → ∀𝑦(∀𝑦𝜑𝜓)) → (∀𝑦𝜑 → ∀𝑦𝜓))
128, 11ja 175 1 ((∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-10 2077  ax-11 2091  ax-12 2104
This theorem depends on definitions:  df-bi 199  df-ex 1743
This theorem is referenced by:  axc5c4c711toc5  40095  axc5c4c711toc4  40096  axc5c4c711toc7  40097  axc5c4c711to11  40098
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