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Theorem 19.28v 2074
Description: Version of 19.28 2243 with a dv 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 1947 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.3v 2063 . . 3 (∀𝑥𝜑𝜑)
32anbi1i 733 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
41, 3bitri 264 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wal 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854
This theorem is referenced by:  reu6  3536  dfer2  7912  kmlem14  9177  kmlem15  9178  bnj1176  31380  bnj1186  31382  19.28vv  39087
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