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Theorem 19.38 1832
Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1833 and 19.38b 1834. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2199. (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 1775 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 123 . . . 4 𝜑 → (𝜑𝜓))
32alimi 1805 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
41, 3sylbir 236 . 2 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
5 ala1 1807 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
64, 5ja 187 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 208  df-ex 1774
This theorem is referenced by:  19.38a  1833  19.38b  1834  nfimd  1888  19.21v  1933  19.23v  1936  bj-nfimexal  33845  bj-nfimt  33857  bj-wnf1  33937  bj-nnfim  33961  bj-19.21t  33984  bj-19.23t  33985  bj-19.21t0  34039  pm10.53  40565
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