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Theorem wl-ax11-lem8 33906
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem8 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))
Distinct variable group:   𝑥,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢)

Proof of Theorem wl-ax11-lem8
StepHypRef Expression
1 axc11n 2447 . . 3 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
21con3i 152 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
3 wl-ax11-lem1 33899 . . . . . . 7 (∀𝑢 𝑢 = 𝑦 → (∀𝑢 𝑢 = 𝑥 ↔ ∀𝑦 𝑦 = 𝑥))
43notbid 310 . . . . . 6 (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑢 𝑢 = 𝑥 ↔ ¬ ∀𝑦 𝑦 = 𝑥))
54anbi1d 623 . . . . 5 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑)))
64anbi1d 623 . . . . . . . 8 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑)))
7 axc11n 2447 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
87con3i 152 . . . . . . . . . 10 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑦)
9 wl-ax11-lem4 33902 . . . . . . . . . . . 12 𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
10 sbequ12 2286 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
1110equcoms 2124 . . . . . . . . . . . . . 14 (𝑢 = 𝑦 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
1211sps 2226 . . . . . . . . . . . . 13 (∀𝑢 𝑢 = 𝑦 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
1312adantr 474 . . . . . . . . . . . 12 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
149, 13albid 2265 . . . . . . . . . . 11 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑦]𝜑))
1514ex 403 . . . . . . . . . 10 (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑦]𝜑)))
168, 15syl5 34 . . . . . . . . 9 (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑦]𝜑)))
1716pm5.32d 572 . . . . . . . 8 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑)))
186, 17bitr4d 274 . . . . . . 7 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑)))
1918dral1 2460 . . . . . 6 (∀𝑢 𝑢 = 𝑦 → (∀𝑢(¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ ∀𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑)))
20 wl-ax11-lem7 33905 . . . . . 6 (∀𝑢(¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑))
21 wl-ax11-lem7 33905 . . . . . 6 (∀𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦𝑥𝜑))
2219, 20, 213bitr3g 305 . . . . 5 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦𝑥𝜑)))
235, 22bitr3d 273 . . . 4 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦𝑥𝜑)))
24 pm5.32 569 . . . 4 ((¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑)) ↔ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦𝑥𝜑)))
2523, 24sylibr 226 . . 3 (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑)))
2625imp 397 . 2 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))
272, 26sylan2 586 1 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wal 1654  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-12 2220  ax-13 2389  ax-wl-11v 33898
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068
This theorem is referenced by:  wl-ax11-lem10  33908
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