Theorem axc5c4c711toc4 42021

Description: Rederivation of axc4 2315 from axc5c4c711 42019. Note that only propositional calculus is required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc5c4c711toc4	⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem axc5c4c711toc4

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (𝜑 → ∀𝑥(∀𝑥𝜑 → 𝜓)))
2		ax-1 6	. 2 ⊢ ((𝜑 → ∀𝑥(∀𝑥𝜑 → 𝜓)) → (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥𝜑 → 𝜓) → (𝜑 → ∀𝑥(∀𝑥𝜑 → 𝜓))))
3		axc5c4c711 42019	. 2 ⊢ ((∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥𝜑 → 𝜓) → (𝜑 → ∀𝑥(∀𝑥𝜑 → 𝜓))) → (∀𝑥𝜑 → ∀𝑥𝜓))
4	1, 2, 3	3syl 18	1 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)