Theorem xfree2 33119

Description: A partial converse to 19.9t 2244. (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 33118	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
2		eximal 1815	. . 3 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	2	albii 1852	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
4	1, 3	bitri 278	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by: (None)