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Theorem 19.32v 1868
 Description: Version of 19.32 2100 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
Assertion
Ref Expression
19.32v (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.32v
StepHypRef Expression
1 19.21v 1867 . 2 (∀𝑥𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
2 df-or 385 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32albii 1746 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥𝜑𝜓))
4 df-or 385 . 2 ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
51, 3, 43bitr4i 292 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383  ∀wal 1480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838 This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1704 This theorem is referenced by:  19.31v  1869  pm10.12  38383
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