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Theorem axc5c4c711toc5 37625
 Description: Rederivation of sp 2041 from axc5c4c711 37624. Note that ax6 2239 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1876 instead of ax6 2239, so that this rederivation requires only ax6v 1876 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 1876 . . 3 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 pm2.21 119 . . . 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 37624 . . . 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 189 . 2 𝜑 → ¬ ∀𝑥𝜑)
76con4i 112 1 (∀𝑥𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473   = wceq 1475 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by: (None)
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