Theorem wl-ax11-lem8 34818

Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem8	⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥𝜑))

Distinct variable group:   𝑥,𝑢

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢)

Proof of Theorem wl-ax11-lem8

Step	Hyp	Ref	Expression
1		axc11n 2444	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
2	1	con3i 157	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
3		wl-ax11-lem1 34811	. . . . . . 7 ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢 𝑢 = 𝑥 ↔ ∀𝑦 𝑦 = 𝑥))
4	3	notbid 320	. . . . . 6 ⊢ (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑢 𝑢 = 𝑥 ↔ ¬ ∀𝑦 𝑦 = 𝑥))
5	4	anbi1d 631	. . . . 5 ⊢ (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑)))
6	4	anbi1d 631	. . . . . . . 8 ⊢ (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑)))
7		axc11n 2444	. . . . . . . . . . 11 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
8	7	con3i 157	. . . . . . . . . 10 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑦)
9		wl-ax11-lem4 34814	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
10		sbequ12 2249	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
11	10	equcoms 2023	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑦 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
12	11	sps 2180	. . . . . . . . . . . . 13 ⊢ (∀𝑢 𝑢 = 𝑦 → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
13	12	adantr 483	. . . . . . . . . . . 12 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝜑 ↔ [𝑢 / 𝑦]𝜑))
14	9, 13	albid 2220	. . . . . . . . . . 11 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑦]𝜑))
15	14	ex 415	. . . . . . . . . 10 ⊢ (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑦]𝜑)))
16	8, 15	syl5 34	. . . . . . . . 9 ⊢ (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑦]𝜑)))
17	16	pm5.32d 579	. . . . . . . 8 ⊢ (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑)))
18	6, 17	bitr4d 284	. . . . . . 7 ⊢ (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑)))
19	18	dral1 2457	. . . . . 6 ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢(¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ ∀𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑)))
20		wl-ax11-lem7 34817	. . . . . 6 ⊢ (∀𝑢(¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑))
21		wl-ax11-lem7 34817	. . . . . 6 ⊢ (∀𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑥𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦∀𝑥𝜑))
22	19, 20, 21	3bitr3g 315	. . . . 5 ⊢ (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑢 𝑢 = 𝑥 ∧ ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦∀𝑥𝜑)))
23	5, 22	bitr3d 283	. . . 4 ⊢ (∀𝑢 𝑢 = 𝑦 → ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦∀𝑥𝜑)))
24		pm5.32 576	. . . 4 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥𝜑)) ↔ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑) ↔ (¬ ∀𝑦 𝑦 = 𝑥 ∧ ∀𝑦∀𝑥𝜑)))
25	23, 24	sylibr 236	. . 3 ⊢ (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥𝜑)))
26	25	imp 409	. 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑥) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥𝜑))
27	2, 26	sylan2 594	1 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2173  ax-13 2386  ax-wl-11v 34810

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066

This theorem is referenced by:  wl-ax11-lem10  34820