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Theorem sbal2 2134
 Description: Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
sbal2 x x = y → ([z / y]xφx[z / y]φ))
Distinct variable groups:   y,z   x,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbal2
StepHypRef Expression
1 alcom 1737 . . 3 (yx(y = zφ) ↔ xy(y = zφ))
2 nfnae 1956 . . . 4 y ¬ x x = y
3 nfnae 1956 . . . . . 6 x ¬ x x = y
4 dveeq1 2018 . . . . . 6 x x = y → (y = zx y = z))
53, 4nfd 1766 . . . . 5 x x = y → Ⅎx y = z)
6 19.21t 1795 . . . . 5 (Ⅎx y = z → (x(y = zφ) ↔ (y = zxφ)))
75, 6syl 15 . . . 4 x x = y → (x(y = zφ) ↔ (y = zxφ)))
82, 7albid 1772 . . 3 x x = y → (yx(y = zφ) ↔ y(y = zxφ)))
91, 8syl5rbbr 251 . 2 x x = y → (y(y = zxφ) ↔ xy(y = zφ)))
10 sb6 2099 . 2 ([z / y]xφy(y = zxφ))
11 sb6 2099 . . 3 ([z / y]φy(y = zφ))
1211albii 1566 . 2 (x[z / y]φxy(y = zφ))
139, 10, 123bitr4g 279 1 x x = y → ([z / y]xφx[z / y]φ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544  [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: (None)
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