Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-nfalt Structured version   Visualization version   GIF version

Theorem bj-nfalt 31889
 Description: Closed form of nfal 2139. (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-nfalt (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Proof of Theorem bj-nfalt
StepHypRef Expression
1 bj-hbalt 31858 . . . 4 (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
21alimi 1730 . . 3 (∀𝑦𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
32alcoms 2022 . 2 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
4 nf5 2102 . . 3 (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
54albii 1737 . 2 (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦(𝜑 → ∀𝑦𝜑))
6 nf5 2102 . 2 (Ⅎ𝑦𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
73, 5, 63imtr4i 280 1 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  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-10 2006  ax-11 2021  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:  bj-dvelimdv1  32028
 Copyright terms: Public domain W3C validator