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Theorem eujust 2206
 Description: A soundness justification theorem for df-eu 2208, showing that the definition is equivalent to itself with its dummy variable renamed. Note that y and z needn't be distinct variables. See eujustALT 2207 for a proof that provides an example of how it can be achieved through the use of dvelim 2016. (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 1686 . . . . 5 (y = w → (x = yx = w))
21bibi2d 309 . . . 4 (y = w → ((φx = y) ↔ (φx = w)))
32albidv 1625 . . 3 (y = w → (x(φx = y) ↔ x(φx = w)))
43cbvexv 2003 . 2 (yx(φx = y) ↔ wx(φx = w))
5 equequ2 1686 . . . . 5 (w = z → (x = wx = z))
65bibi2d 309 . . . 4 (w = z → ((φx = w) ↔ (φx = z)))
76albidv 1625 . . 3 (w = z → (x(φx = w) ↔ x(φx = z)))
87cbvexv 2003 . 2 (wx(φx = w) ↔ zx(φx = z))
94, 8bitri 240 1 (yx(φx = y) ↔ zx(φx = z))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642 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 This theorem is referenced by: (None)
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