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Theorem 19.38 1834
Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under nonfreeness conditions, see 19.38a 1835 and 19.38b 1836. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2195. (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 1776 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 123 . . . 4 𝜑 → (𝜑𝜓))
32alimi 1806 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
41, 3sylbir 234 . 2 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
5 ala1 1808 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
64, 5ja 186 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804
This theorem depends on definitions:  df-bi 206  df-ex 1775
This theorem is referenced by:  19.38a  1835  19.38b  1836  nfimd  1890  19.21v  1935  19.23v  1938  bj-nfimexal  36102  bj-nfimt  36114  bj-wnf1  36194  bj-substax12  36198  bj-nnfim  36223  bj-19.21t  36246  bj-19.23t  36247  bj-19.21t0  36307  pm10.53  43803
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