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Theorem 19.21t 2204
 Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2205. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
Assertion
Ref Expression
19.21t (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21t
StepHypRef Expression
1 19.38a 1841 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
2 19.9t 2202 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
32imbi1d 345 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
41, 3bitr3d 284 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786 This theorem is referenced by:  19.21  2205  sbal1  2548  sbal2  2549  sbal2OLD  2550  r19.21t  3178  ceqsalt  3475  sbciegft  3758  bj-ceqsalt0  34475  bj-ceqsalt1  34476  wl-sbhbt  35106  wl-2sb6d  35110  wl-sbalnae  35114  wl-dfralf  35155  ax12indalem  36392  ax12inda2ALT  36393
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