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Theorem 19.38 1841
 Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1842 and 19.38b 1843. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2205. (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 1784 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 123 . . . 4 𝜑 → (𝜑𝜓))
32alimi 1814 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
41, 3sylbir 238 . 2 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
5 ala1 1816 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
64, 5ja 189 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812 This theorem depends on definitions:  df-bi 210  df-ex 1783 This theorem is referenced by:  19.38a  1842  19.38b  1843  nfimd  1896  19.21v  1941  19.23v  1944  bj-nfimexal  34346  bj-nfimt  34358  bj-wnf1  34438  bj-subst  34442  bj-nnfim  34464  bj-19.21t  34487  bj-19.23t  34488  bj-19.21t0  34542  pm10.53  41436
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