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Theorem sbnv 1765
 Description: Version of sbn 1823 where x and y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
Assertion
Ref Expression
sbnv ([y / x] ¬ φ ↔ ¬ [y / x]φ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem sbnv
StepHypRef Expression
1 sb6 1763 . . 3 ([y / x] ¬ φx(x = y → ¬ φ))
2 alinexa 1491 . . 3 (x(x = y → ¬ φ) ↔ ¬ x(x = y φ))
31, 2bitri 173 . 2 ([y / x] ¬ φ ↔ ¬ x(x = y φ))
4 sb5 1764 . 2 ([y / x]φx(x = y φ))
53, 4xchbinxr 607 1 ([y / x] ¬ φ ↔ ¬ [y / x]φ)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-sb 1643 This theorem is referenced by:  sbn  1823
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