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Theorem 19.38 1806
 Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1807 and 19.38b 1808. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2111. (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 1746 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 120 . . . 4 𝜑 → (𝜑𝜓))
32alimi 1779 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
41, 3sylbir 225 . 2 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
5 ala1 1781 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
64, 5ja 173 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777 This theorem depends on definitions:  df-bi 197  df-ex 1745 This theorem is referenced by:  19.38a  1807  19.38b  1808  nfimt  1861  19.21v  1908  19.23v  1911  19.23vOLD  1959  19.21tOLDOLD  2112  19.21tOLD  2249  bj-19.21t  32942  pm10.53  38882
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