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Theorem 19.31v 2022
Description: Version of 19.31 2258 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
Assertion
Ref Expression
19.31v (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.31v
StepHypRef Expression
1 19.32v 2021 . 2 (∀𝑥(𝜓𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑))
2 orcom 857 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
32albii 1895 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜓𝜑))
4 orcom 857 . 2 ((∀𝑥𝜑𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
51, 3, 43bitr4i 292 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 834  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991
This theorem depends on definitions:  df-bi 197  df-or 835  df-ex 1853
This theorem is referenced by:  19.31vv  39109
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