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Theorem 19.38 1934
Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1935 and 19.38b 1937. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2240. (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 1877 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 121 . . . 4 𝜑 → (𝜑𝜓))
32alimi 1907 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
41, 3sylbir 227 . 2 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
5 ala1 1909 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
64, 5ja 175 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1651  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905
This theorem depends on definitions:  df-bi 199  df-ex 1876
This theorem is referenced by:  19.38a  1935  19.38aOLD  1936  19.38b  1937  19.38bOLD  1938  nfimd  1993  19.21v  2035  19.23v  2038  19.23vOLD  2086  bj-nfimt  33122  bj-19.21t  33312  pm10.53  39343
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