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Theorem wl-ax11-lem6 35741
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 35735 . . 3 (∀𝑢𝑥[𝑢 / 𝑦]𝜑 → ∀𝑥𝑢[𝑢 / 𝑦]𝜑)
2 ax-wl-11v 35735 . . 3 (∀𝑥𝑢[𝑢 / 𝑦]𝜑 → ∀𝑢𝑥[𝑢 / 𝑦]𝜑)
31, 2impbii 208 . 2 (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑢[𝑢 / 𝑦]𝜑)
4 nfna1 2149 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
5 wl-ax11-lem3 35738 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)
64, 5nfan1 2193 . . . 4 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦)
7 wl-ax11-lem5 35740 . . . . 5 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
87adantl 482 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
96, 8albid 2215 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑥𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
109ancoms 459 . 2 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
113, 10bitrid 282 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 205  wa 396  wal 1537  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372  ax-wl-11v 35735
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068
This theorem is referenced by:  wl-ax11-lem10  35745
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