Theorem xfree 32267

Description: A partial converse to 19.9t 2193. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)

Assertion

Ref	Expression
xfree	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Proof of Theorem xfree

Step	Hyp	Ref	Expression
1		nf5 2272	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nf6 2273	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
3	1, 2	bitr3i 277	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532  ∃wex 1774  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-or 847  df-ex 1775  df-nf 1779

This theorem is referenced by:  xfree2  32268