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Theorem 19.38 1756
Description: Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2060. (Revised by Wolf Lammen, 2-Jan-2018.)
Assertion
Ref Expression
19.38 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Proof of Theorem 19.38
StepHypRef Expression
1 alnex 1696 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 118 . . . 4 𝜑 → (𝜑𝜓))
32alimi 1729 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
41, 3sylbir 223 . 2 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
5 ala1 1754 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
64, 5ja 171 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by:  19.21v  1854  19.23v  1888  19.21t  2060  19.21tOLD  2200  bj-19.21t  31798  wl-nf2-nf  32247  pm10.53  37370
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