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Theorem 19.28v 2039
Description: Version of 19.28 2213 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 1916 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.3v 2031 . . 3 (∀𝑥𝜑𝜑)
32anbi1i 617 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
41, 3bitri 267 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  wal 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824
This theorem is referenced by:  reu6  3606  dfer2  8027  kmlem14  9320  kmlem15  9321  bnj1176  31686  bnj1186  31688  19.28vv  39533
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