Theorem xfree2 30807

Description: A partial converse to 19.9t 2197. (Contributed by Stefan Allan, 21-Dec-2008.)

Assertion

Ref	Expression
xfree2	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem xfree2

Step	Hyp	Ref	Expression
1		xfree 30806	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
2		eximal 1785	. . 3 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	2	albii 1822	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
4	1, 3	bitri 274	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782

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-12 2171

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by: (None)