Theorem xfree 28687

Description: A partial converse to 19.9t 2059. (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 2102	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nf6 2103	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
3	1, 2	bitr3i 265	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by:  xfree2  28688