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Theorem 19.23t 2066
 Description: Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2067. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.)
Assertion
Ref Expression
19.23t (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Proof of Theorem 19.23t
StepHypRef Expression
1 nfnt 1767 . . 3 (Ⅎ𝑥𝜓 → Ⅎ𝑥 ¬ 𝜓)
2 19.21t 2061 . . 3 (Ⅎ𝑥 ¬ 𝜓 → (∀𝑥𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)))
31, 2syl 17 . 2 (Ⅎ𝑥𝜓 → (∀𝑥𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)))
4 con34b 305 . . 3 ((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
54albii 1737 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥𝜓 → ¬ 𝜑))
6 eximal 1698 . 2 ((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
73, 5, 63bitr4g 302 1 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by:  19.23  2067  axie2  2585  r19.23t  3003  ceqsalt  3201  vtoclgft  3227  vtoclgftOLD  3228  sbciegft  3433  bj-ceqsalt0  32067  bj-ceqsalt1  32068  wl-equsald  32504
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