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Theorem sbn 2062
 Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbn ([y / x] ¬ φ ↔ ¬ [y / x]φ)

Proof of Theorem sbn
StepHypRef Expression
1 sbequ2 1650 . . . . 5 (x = y → ([y / x] ¬ φ → ¬ φ))
2 sbequ2 1650 . . . . 5 (x = y → ([y / x]φφ))
31, 2nsyld 132 . . . 4 (x = y → ([y / x] ¬ φ → ¬ [y / x]φ))
43sps 1754 . . 3 (x x = y → ([y / x] ¬ φ → ¬ [y / x]φ))
5 sb4 2053 . . . 4 x x = y → ([y / x] ¬ φx(x = y → ¬ φ)))
6 sb1 1651 . . . . . 6 ([y / x]φx(x = y φ))
7 equs3 1644 . . . . . 6 (x(x = y φ) ↔ ¬ x(x = y → ¬ φ))
86, 7sylib 188 . . . . 5 ([y / x]φ → ¬ x(x = y → ¬ φ))
98con2i 112 . . . 4 (x(x = y → ¬ φ) → ¬ [y / x]φ)
105, 9syl6 29 . . 3 x x = y → ([y / x] ¬ φ → ¬ [y / x]φ))
114, 10pm2.61i 156 . 2 ([y / x] ¬ φ → ¬ [y / x]φ)
12 sbequ1 1918 . . . 4 (x = y → (φ → [y / x]φ))
1312con3rr3 128 . . 3 (¬ [y / x]φ → (x = y → ¬ φ))
14 sb2 2023 . . . . . 6 (x(x = y → ¬ ¬ φ) → [y / x] ¬ ¬ φ)
15 notnot 282 . . . . . . 7 (φ ↔ ¬ ¬ φ)
1615sbbii 1653 . . . . . 6 ([y / x]φ ↔ [y / x] ¬ ¬ φ)
1714, 16sylibr 203 . . . . 5 (x(x = y → ¬ ¬ φ) → [y / x]φ)
1817con3i 127 . . . 4 (¬ [y / x]φ → ¬ x(x = y → ¬ ¬ φ))
19 equs3 1644 . . . 4 (x(x = y ¬ φ) ↔ ¬ x(x = y → ¬ ¬ φ))
2018, 19sylibr 203 . . 3 (¬ [y / x]φx(x = y ¬ φ))
21 df-sb 1649 . . 3 ([y / x] ¬ φ ↔ ((x = y → ¬ φ) x(x = y ¬ φ)))
2213, 20, 21sylanbrc 645 . 2 (¬ [y / x]φ → [y / x] ¬ φ)
2311, 22impbii 180 1 ([y / x] ¬ φ ↔ ¬ [y / x]φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  [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:  sbi2  2064  sbor  2066  sban  2069  spsbe  2075  sb8e  2093  sbex  2128  sbcng  3086  difab  3523  complab  3524
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