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Theorem wl-ax11-lem8 33001
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 2306 . . 3 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
21con3i 150 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
3 wl-ax11-lem1 32994 . . . . . . 7 (∀𝑢 𝑢 = 𝑦 → (∀𝑢 𝑢 = 𝑥 ↔ ∀𝑦 𝑦 = 𝑥))
43notbid 308 . . . . . 6 (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑢 𝑢 = 𝑥 ↔ ¬ ∀𝑦 𝑦 = 𝑥))
54anbi1d 740 . . . . 5 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑)))
64anbi1d 740 . . . . . . . 8 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑)))
7 axc11n 2306 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
87con3i 150 . . . . . . . . . 10 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑦)
9 wl-ax11-lem4 32997 . . . . . . . . . . . 12 𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
10 sbequ12 2108 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
1110equcoms 1944 . . . . . . . . . . . . . 14 (𝑢 = 𝑦 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
1211sps 2053 . . . . . . . . . . . . 13 (∀𝑢 𝑢 = 𝑦 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
1312adantr 481 . . . . . . . . . . . 12 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
149, 13albid 2088 . . . . . . . . . . 11 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑦]𝜑))
1514ex 450 . . . . . . . . . 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 670 . . . . . . . 8 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑)))
186, 17bitr4d 271 . . . . . . 7 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑)))
1918dral1 2324 . . . . . 6 (∀𝑢 𝑢 = 𝑦 → (∀𝑢(¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ ∀𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑)))
20 wl-ax11-lem7 33000 . . . . . 6 (∀𝑢(¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑))
21 wl-ax11-lem7 33000 . . . . . 6 (∀𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦𝑥𝜑))
2219, 20, 213bitr3g 302 . . . . 5 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦𝑥𝜑)))
235, 22bitr3d 270 . . . 4 (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦𝑥𝜑)))
24 pm5.32 667 . . . 4 ((¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑)) ↔ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦𝑥𝜑)))
2523, 24sylibr 224 . . 3 (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑)))
2625imp 445 . 2 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))
272, 26sylan2 491 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 196  wa 384  wal 1478  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245  ax-wl-11v 32993
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  wl-ax11-lem10  33003
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