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

Theorem 19.21t 2061
 Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2062. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.)
Assertion
Ref Expression
19.21t (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21t
StepHypRef Expression
1 nf5r 2052 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2 alim 1729 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
31, 2syl9 75 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
4 19.9t 2059 . . . 4 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
54imbi1d 330 . . 3 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
6 19.38 1757 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
75, 6syl6bir 243 . 2 (Ⅎ𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓)))
83, 7impbid 201 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → 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-ex 1696  df-nf 1701 This theorem is referenced by:  19.21  2062  stdpc5OLD  2064  19.23t  2066  sbal1  2448  sbal2  2449  r19.21t  2938  ceqsalt  3201  sbciegft  3433  bj-ceqsalt0  32067  bj-ceqsalt1  32068  wl-sbhbt  32514  wl-2sb6d  32520  wl-sbalnae  32524  ax12indalem  33248  ax12inda2ALT  33249
 Copyright terms: Public domain W3C validator