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Theorem 19.38 1830
Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1831 and 19.38b 1832. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2196. (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 1773 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 123 . . . 4 𝜑 → (𝜑𝜓))
32alimi 1803 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
41, 3sylbir 236 . 2 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
5 ala1 1805 . 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 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-ex 1772
This theorem is referenced by:  19.38a  1831  19.38b  1832  nfimd  1886  19.21v  1931  19.23v  1934  bj-nfimexal  33856  bj-nfimt  33868  bj-wnf1  33948  bj-nnfim  33972  bj-19.21t  33995  bj-19.23t  33996  bj-19.21t0  34050  pm10.53  40575
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