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Theorem wl-ax11-lem6 37110
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 37104 . . 3 (∀𝑢𝑥[𝑢 / 𝑦]𝜑 → ∀𝑥𝑢[𝑢 / 𝑦]𝜑)
2 ax-wl-11v 37104 . . 3 (∀𝑥𝑢[𝑢 / 𝑦]𝜑 → ∀𝑢𝑥[𝑢 / 𝑦]𝜑)
31, 2impbii 208 . 2 (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑢[𝑢 / 𝑦]𝜑)
4 nfna1 2141 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
5 wl-ax11-lem3 37107 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)
64, 5nfan1 2188 . . . 4 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦)
7 wl-ax11-lem5 37109 . . . . 5 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
87adantl 480 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
96, 8albid 2210 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑥𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
109ancoms 457 . 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 394  wal 1531  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2166  ax-13 2365  ax-wl-11v 37104
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060
This theorem is referenced by:  wl-ax11-lem10  37114
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