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Theorem axc5c4c711toc5 39572
 Description: Rederivation of sp 2167 from axc5c4c711 39571. Note that ax6 2348 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 2022 instead of ax6 2348, so that this rederivation requires only ax6v 2022 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c4c711toc5 (∀𝑥𝜑𝜑)

Proof of Theorem axc5c4c711toc5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax6v 2022 . . 3 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 pm2.21 121 . . . 4 𝜑 → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦)))
3 ax-1 6 . . . 4 ((𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦)) → (∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦))))
4 axc5c4c711 39571 . . . 4 ((∀𝑥𝑥 ¬ ∀𝑥𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦))) → (∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
52, 3, 43syl 18 . . 3 𝜑 → (∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
61, 5mtoi 191 . 2 𝜑 → ¬ ∀𝑥𝜑)
76con4i 114 1 (∀𝑥𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599   = wceq 1601 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163 This theorem depends on definitions:  df-bi 199  df-ex 1824 This theorem is referenced by: (None)
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