Theorem sbal1 2532

Description: Check out sbal 2163 for a version not dependent on ax-13 2371. A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑧 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbal1	⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbal1

Step	Hyp	Ref	Expression
1		sb4b 2474	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
2		nfnae 2433	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑧
3		nfeqf2 2376	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧)
4		19.21t 2204	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑦 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
5	4	bicomd 226	. . . . . . 7 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑥(𝑦 = 𝑧 → 𝜑)))
6	3, 5	syl 17	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑥(𝑦 = 𝑧 → 𝜑)))
7	2, 6	albid 2220	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
8	1, 7	sylan9bbr 514	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
9		nfnae 2433	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
10		sb4b 2474	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜑)))
11	9, 10	albid 2220	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑)))
12		alcom 2160	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑))
13	11, 12	bitrdi 290	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
14	13	adantl 485	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
15	8, 14	bitr4d 285	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
16	15	ex 416	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)))
17		sbequ12 2249	. . . 4 ⊢ (𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
18	17	sps 2182	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
19		sbequ12 2249	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
20	19	sps 2182	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
21	20	dral2 2437	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
22	18, 21	bitr3d 284	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
23	16, 22	pm2.61d2 184	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  Ⅎwnf 1791  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071

This theorem is referenced by: (None)