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Theorem 19.23t 2078
Description: Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2079. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1709 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 1781 . . 3 (Ⅎ𝑥𝜓 → Ⅎ𝑥 ¬ 𝜓)
2 19.21t 2072 . . 3 (Ⅎ𝑥 ¬ 𝜓 → (∀𝑥𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)))
31, 2syl 17 . 2 (Ⅎ𝑥𝜓 → (∀𝑥𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)))
4 con34b 306 . . 3 ((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
54albii 1746 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥𝜓 → ¬ 𝜑))
6 eximal 1706 . 2 ((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
73, 5, 63bitr4g 303 1 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1480  wex 1703  wnf 1707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1704  df-nf 1709
This theorem is referenced by:  19.23  2079  axie2  2596  r19.23t  3019  ceqsalt  3226  vtoclgft  3252  vtoclgftOLD  3253  sbciegft  3464  bj-ceqsalt0  32857  bj-ceqsalt1  32858  wl-equsald  33305
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