Theorem bj-biexal1 37822

Description: A general FOL biconditional that generalizes 19.9ht 2354 among others. For this and the following theorems, see also 19.35 1912, 19.21 2247, 19.23 2251. 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 2191	. 2 ⊢ Ⅎ𝑥∀𝑥𝜓
2	1	19.23 2251	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  bj-biexal3  37824