Theorem bj-biexal2 34888

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

Assertion

Ref	Expression
bj-biexal2	⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-biexal2

Step	Hyp	Ref	Expression
1		nfe1 2147	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	19.21 2200	1 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by:  bj-biexal3  34889