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Theorem 19.38 1913
 Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1914 and 19.38b 1916. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2228. (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 1853 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 121 . . . 4 𝜑 → (𝜑𝜓))
32alimi 1886 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
41, 3sylbir 225 . 2 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
5 ala1 1888 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
64, 5ja 174 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1628  ∃wex 1851 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884 This theorem depends on definitions:  df-bi 197  df-ex 1852 This theorem is referenced by:  19.38a  1914  19.38aOLD  1915  19.38b  1916  19.38bOLD  1917  nfimd  1972  19.21v  2019  19.23v  2022  19.23vOLD  2070  19.21tOLDOLD  2229  19.21tOLD  2374  bj-nfimt  32948  bj-19.21t  33146  pm10.53  39084
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