Theorem xfree 32762

Description: A partial converse to 19.9t 2238. (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 2315	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nf6 2316	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
3	1, 2	bitr3i 280	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1566  ∃wex 1807  Ⅎwnf 1811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-10 2174  ax-12 2211

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1808  df-nf 1812

This theorem is referenced by:  xfree2  32763