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Theorem 19.28v 1998
Description: Version of 19.28 2225 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
19.28v (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.28v
StepHypRef Expression
1 19.26 1877 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.3v 1989 . . 3 (∀𝑥𝜑𝜑)
32anbi1i 624 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
41, 3bitri 274 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787
This theorem is referenced by:  reu6  3665  dfer2  8491  kmlem14  9930  kmlem15  9931  bnj1176  32994  bnj1186  32996  ismnuprim  41894  19.28vv  41986
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