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Theorem 19.28v 1907
Description: Version of 19.28 2094 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 1796 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.3v 1895 . . 3 (∀𝑥𝜑𝜑)
32anbi1i 730 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
41, 3bitri 264 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wal 1479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by:  reu6  3389  dfer2  7728  kmlem14  8970  kmlem15  8971  bnj1176  31047  bnj1186  31049  19.28vv  38405
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