Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hbexg Structured version   Visualization version   GIF version

Theorem hbexg 37793
 Description: Closed form of nfex 2140. Derived from hbexgVD 38164. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbexg (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Proof of Theorem hbexg
StepHypRef Expression
1 nfa2 2027 . . 3 𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑)
2 sp 2041 . . . . . . 7 (∀𝑦(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
32alimi 1730 . . . . . 6 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
4 nf5 2102 . . . . . 6 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
53, 4sylibr 223 . . . . 5 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
61, 5nfexd 2153 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝑦𝜑)
7 nf5 2102 . . . 4 (Ⅎ𝑥𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
86, 7sylib 207 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
91, 8alrimi 2069 . 2 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
10 alcom 2024 . 2 (∀𝑦𝑥(∃𝑦𝜑 → ∀𝑥𝑦𝜑) ↔ ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
119, 10sylib 207 1 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀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-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: (None)
 Copyright terms: Public domain W3C validator