Theorem wl-sbalnae 33345

Description: A theorem used in elimination of disjoint variable restrictions by replacing them with distinctors. (Contributed by Wolf Lammen, 25-Jul-2019.)

Assertion

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

Proof of Theorem wl-sbalnae

Step	Hyp	Ref	Expression
1		sb4b 2358	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
2		nfnae 2318	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
3		nfnae 2318	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑧
4	2, 3	nfan 1828	. . . . . 6 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
5		nfeqf 2301	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
6		19.21t 2073	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑦 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
7	6	bicomd 213	. . . . . . 7 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑥(𝑦 = 𝑧 → 𝜑)))
8	5, 7	syl 17	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑥(𝑦 = 𝑧 → 𝜑)))
9	4, 8	albid 2090	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
10	1, 9	sylan9bbr 737	. . . 4 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
11		nfnae 2318	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
12		sb4b 2358	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜑)))
13	11, 12	albid 2090	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑)))
14		alcom 2037	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑))
15	13, 14	syl6bb 276	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
16	15	adantl 482	. . . 4 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
17	10, 16	bitr4d 271	. . 3 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
18	17	ex 450	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)))
19		sbequ12 2111	. . . 4 ⊢ (𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
20	19	sps 2055	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
21		sbequ12 2111	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
22	21	sps 2055	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
23	22	dral2 2324	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
24	20, 23	bitr3d 270	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
25	18, 24	pm2.61d2 172	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  Ⅎwnf 1708  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  wl-sbal1  33346  wl-sbal2  33347