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Theorem 19.28 2217
Description: Theorem 19.28 of [Margaris] p. 90. See 19.28v 1987 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 7-May-2025.)
Hypothesis
Ref Expression
19.28.1 𝑥𝜑
Assertion
Ref Expression
19.28 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.28
StepHypRef Expression
1 19.26 1866 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.28.1 . . 3 𝑥𝜑
3219.3 2191 . 2 (∀𝑥𝜑𝜑)
41, 3bianbi 625 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wal 1532  wnf 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-nf 1779
This theorem is referenced by:  aaanOLD  2323  wl-ax11-lem7  37299
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