Theorem bj-biexal1 36718

Description: A general FOL biconditional that generalizes 19.9ht 2320 among others. For this and the following theorems, see also 19.35 1878, 19.21 2209, 19.23 2213. When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-biexal1	⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-biexal1

Step	Hyp	Ref	Expression
1		nfa1 2153	. 2 ⊢ Ⅎ𝑥∀𝑥𝜓
2	1	19.23 2213	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2143  ax-12 2179

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1781  df-nf 1785

This theorem is referenced by:  bj-biexal3  36720