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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]xφx[z / y]φ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbal1
StepHypRef Expression
1 sbequ12 1919 . . . . 5 (y = z → (xφ ↔ [z / y]xφ))
21sps 1754 . . . 4 (y y = z → (xφ ↔ [z / y]xφ))
3 sbequ12 1919 . . . . . 6 (y = z → (φ ↔ [z / y]φ))
43sps 1754 . . . . 5 (y y = z → (φ ↔ [z / y]φ))
54dral2 1966 . . . 4 (y y = z → (xφx[z / y]φ))
62, 5bitr3d 246 . . 3 (y y = z → ([z / y]xφx[z / y]φ))
76a1d 22 . 2 (y y = z → (¬ x x = z → ([z / y]xφx[z / y]φ)))
8 nfa1 1788 . . . . . . . 8 xxφ
98nfsb4 2081 . . . . . . 7 x x = z → Ⅎx[z / y]xφ)
109nfrd 1763 . . . . . 6 x x = z → ([z / y]xφx[z / y]xφ))
11 sp 1747 . . . . . . . 8 (xφφ)
1211sbimi 1652 . . . . . . 7 ([z / y]xφ → [z / y]φ)
1312alimi 1559 . . . . . 6 (x[z / y]xφx[z / y]φ)
1410, 13syl6 29 . . . . 5 x x = z → ([z / y]xφx[z / y]φ))
1514adantl 452 . . . 4 ((¬ y y = z ¬ x x = z) → ([z / y]xφx[z / y]φ))
16 sb4 2053 . . . . . . . 8 y y = z → ([z / y]φy(y = zφ)))
1716al2imi 1561 . . . . . . 7 (x ¬ y y = z → (x[z / y]φxy(y = zφ)))
1817hbnaes 1957 . . . . . 6 y y = z → (x[z / y]φxy(y = zφ)))
19 ax-7 1734 . . . . . 6 (xy(y = zφ) → yx(y = zφ))
2018, 19syl6 29 . . . . 5 y y = z → (x[z / y]φyx(y = zφ)))
21 dveeq2 1940 . . . . . . . . 9 x x = z → (y = zx y = z))
22 alim 1558 . . . . . . . . 9 (x(y = zφ) → (x y = zxφ))
2321, 22syl9 66 . . . . . . . 8 x x = z → (x(y = zφ) → (y = zxφ)))
2423al2imi 1561 . . . . . . 7 (y ¬ x x = z → (yx(y = zφ) → y(y = zxφ)))
25 sb2 2023 . . . . . . 7 (y(y = zxφ) → [z / y]xφ)
2624, 25syl6 29 . . . . . 6 (y ¬ x x = z → (yx(y = zφ) → [z / y]xφ))
2726hbnaes 1957 . . . . 5 x x = z → (yx(y = zφ) → [z / y]xφ))
2820, 27sylan9 638 . . . 4 ((¬ y y = z ¬ x x = z) → (x[z / y]φ → [z / y]xφ))
2915, 28impbid 183 . . 3 ((¬ y y = z ¬ x x = z) → ([z / y]xφx[z / y]φ))
3029ex 423 . 2 y y = z → (¬ x x = z → ([z / y]xφx[z / y]φ)))
317, 30pm2.61i 156 1 x x = z → ([z / y]xφx[z / y]φ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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