Theorem sbal1 2126

Description: A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor ¬ ∀xx = z. (Contributed by NM, 5-Aug-1993.)

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		sbequ12 1919	. . . . 5 ⊢ (y = z → (∀xφ ↔ [z / y]∀xφ))
2	1	sps 1754	. . . 4 ⊢ (∀y y = z → (∀xφ ↔ [z / y]∀xφ))
3		sbequ12 1919	. . . . . 6 ⊢ (y = z → (φ ↔ [z / y]φ))
4	3	sps 1754	. . . . 5 ⊢ (∀y y = z → (φ ↔ [z / y]φ))
5	4	dral2 1966	. . . 4 ⊢ (∀y y = z → (∀xφ ↔ ∀x[z / y]φ))
6	2, 5	bitr3d 246	. . 3 ⊢ (∀y y = z → ([z / y]∀xφ ↔ ∀x[z / y]φ))
7	6	a1d 22	. 2 ⊢ (∀y y = z → (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ)))
8		nfa1 1788	. . . . . . . 8 ⊢ Ⅎx∀xφ
9	8	nfsb4 2081	. . . . . . 7 ⊢ (¬ ∀x x = z → Ⅎx[z / y]∀xφ)
10	9	nfrd 1763	. . . . . 6 ⊢ (¬ ∀x x = z → ([z / y]∀xφ → ∀x[z / y]∀xφ))
11		sp 1747	. . . . . . . 8 ⊢ (∀xφ → φ)
12	11	sbimi 1652	. . . . . . 7 ⊢ ([z / y]∀xφ → [z / y]φ)
13	12	alimi 1559	. . . . . 6 ⊢ (∀x[z / y]∀xφ → ∀x[z / y]φ)
14	10, 13	syl6 29	. . . . 5 ⊢ (¬ ∀x x = z → ([z / y]∀xφ → ∀x[z / y]φ))
15	14	adantl 452	. . . 4 ⊢ ((¬ ∀y y = z ∧ ¬ ∀x x = z) → ([z / y]∀xφ → ∀x[z / y]φ))
16		sb4 2053	. . . . . . . 8 ⊢ (¬ ∀y y = z → ([z / y]φ → ∀y(y = z → φ)))
17	16	al2imi 1561	. . . . . . 7 ⊢ (∀x ¬ ∀y y = z → (∀x[z / y]φ → ∀x∀y(y = z → φ)))
18	17	hbnaes 1957	. . . . . 6 ⊢ (¬ ∀y y = z → (∀x[z / y]φ → ∀x∀y(y = z → φ)))
19		ax-7 1734	. . . . . 6 ⊢ (∀x∀y(y = z → φ) → ∀y∀x(y = z → φ))
20	18, 19	syl6 29	. . . . 5 ⊢ (¬ ∀y y = z → (∀x[z / y]φ → ∀y∀x(y = z → φ)))
21		dveeq2 1940	. . . . . . . . 9 ⊢ (¬ ∀x x = z → (y = z → ∀x y = z))
22		alim 1558	. . . . . . . . 9 ⊢ (∀x(y = z → φ) → (∀x y = z → ∀xφ))
23	21, 22	syl9 66	. . . . . . . 8 ⊢ (¬ ∀x x = z → (∀x(y = z → φ) → (y = z → ∀xφ)))
24	23	al2imi 1561	. . . . . . 7 ⊢ (∀y ¬ ∀x x = z → (∀y∀x(y = z → φ) → ∀y(y = z → ∀xφ)))
25		sb2 2023	. . . . . . 7 ⊢ (∀y(y = z → ∀xφ) → [z / y]∀xφ)
26	24, 25	syl6 29	. . . . . 6 ⊢ (∀y ¬ ∀x x = z → (∀y∀x(y = z → φ) → [z / y]∀xφ))
27	26	hbnaes 1957	. . . . 5 ⊢ (¬ ∀x x = z → (∀y∀x(y = z → φ) → [z / y]∀xφ))
28	20, 27	sylan9 638	. . . 4 ⊢ ((¬ ∀y y = z ∧ ¬ ∀x x = z) → (∀x[z / y]φ → [z / y]∀xφ))
29	15, 28	impbid 183	. . 3 ⊢ ((¬ ∀y y = z ∧ ¬ ∀x x = z) → ([z / y]∀xφ ↔ ∀x[z / y]φ))
30	29	ex 423	. 2 ⊢ (¬ ∀y y = z → (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ)))
31	7, 30	pm2.61i 156	1 ⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbal  2127