Theorem bj-biexal1 34814

Description: A general FOL biconditional that generalizes 19.9ht 2318 among others. For this and the following theorems, see also 19.35 1881, 19.21 2203, 19.23 2207. 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 2150	. 2 ⊢ Ⅎ𝑥∀𝑥𝜓
2	1	19.23 2207	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by:  bj-biexal3  34816