Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-ax11-lem6 Structured version   Visualization version   GIF version

Theorem wl-ax11-lem6 35387
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem6 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
Distinct variable group:   𝑥,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢)

Proof of Theorem wl-ax11-lem6
StepHypRef Expression
1 ax-wl-11v 35381 . . 3 (∀𝑢𝑥[𝑢 / 𝑦]𝜑 → ∀𝑥𝑢[𝑢 / 𝑦]𝜑)
2 ax-wl-11v 35381 . . 3 (∀𝑥𝑢[𝑢 / 𝑦]𝜑 → ∀𝑢𝑥[𝑢 / 𝑦]𝜑)
31, 2impbii 212 . 2 (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑢[𝑢 / 𝑦]𝜑)
4 nfna1 2157 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
5 wl-ax11-lem3 35384 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)
64, 5nfan1 2202 . . . 4 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦)
7 wl-ax11-lem5 35386 . . . . 5 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
87adantl 485 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
96, 8albid 2224 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑥𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
109ancoms 462 . 2 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
113, 10syl5bb 286 1 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1540  [wsb 2074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179  ax-13 2373  ax-wl-11v 35381
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075
This theorem is referenced by:  wl-ax11-lem10  35391
  Copyright terms: Public domain W3C validator