Theorem axc5c4c711toc5 44384

Description: Rederivation of sp 2184 from axc5c4c711 44383. Note that ax6 2383 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1968 instead of ax6 2383, so that this rederivation requires only ax6v 1968 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc5c4c711toc5	⊢ (∀𝑥𝜑 → 𝜑)

Proof of Theorem axc5c4c711toc5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6v 1968	. . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦)))
3		ax-1 6	. . . 4 ⊢ ((𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦)) → (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦))))
4		axc5c4c711 44383	. . . 4 ⊢ ((∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦))) → (∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
5	2, 3, 4	3syl 18	. . 3 ⊢ (¬ 𝜑 → (∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
6	1, 5	mtoi 199	. 2 ⊢ (¬ 𝜑 → ¬ ∀𝑥𝜑)
7	6	con4i 114	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538   = wceq 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by: (None)