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Theorem eujust 1899
 Description: A soundness justification theorem for df-eu 1900, showing that the definition is equivalent to itself with its dummy variable renamed. Note that y and z needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
eujust (yx(φx = y) ↔ zx(φx = z))
Distinct variable groups:   x,y   x,z   φ,y   φ,z
Allowed substitution hint:   φ(x)

Proof of Theorem eujust
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 equequ2 1596 . . . . 5 (y = w → (x = yx = w))
21bibi2d 221 . . . 4 (y = w → ((φx = y) ↔ (φx = w)))
32albidv 1702 . . 3 (y = w → (x(φx = y) ↔ x(φx = w)))
43cbvexv 1792 . 2 (yx(φx = y) ↔ wx(φx = w))
5 equequ2 1596 . . . . 5 (w = z → (x = wx = z))
65bibi2d 221 . . . 4 (w = z → ((φx = w) ↔ (φx = z)))
76albidv 1702 . . 3 (w = z → (x(φx = w) ↔ x(φx = z)))
87cbvexv 1792 . 2 (wx(φx = w) ↔ zx(φx = z))
94, 8bitri 173 1 (yx(φx = y) ↔ zx(φx = z))
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378 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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
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