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Theorem bj-biexal1 32391
Description: A general FOL biconditional that generalizes 19.9ht 2139 among others. For this and the following theorems, see also 19.35 1802, 19.21 2073, 19.23 2078. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-biexal1 (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-biexal1
StepHypRef Expression
1 nfa1 2025 . 2 𝑥𝑥𝜓
2119.23 2078 1 (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-biexal3  32393
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