Theorem bj-bialal 37183

Description: When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-bialal	⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-bialal

Step	Hyp	Ref	Expression
1		nfa1 2185	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	19.21 2242	1 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1800  df-nf 1804

This theorem is referenced by: (None)