MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.21t Structured version   Visualization version   GIF version

Theorem 19.21t 2071
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2073. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1707 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 1764 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
2 19.9t 2069 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
32imbi1d 331 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
41, 3bitr3d 270 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707
This theorem is referenced by:  19.21  2073  stdpc5OLD  2075  19.23t  2077  sbal1  2459  sbal2  2460  r19.21t  2951  ceqsalt  3218  sbciegft  3453  bj-ceqsalt0  32573  bj-ceqsalt1  32574  wl-sbhbt  33006  wl-2sb6d  33012  wl-sbalnae  33016  ax12indalem  33749  ax12inda2ALT  33750
  Copyright terms: Public domain W3C validator