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Theorem eujust 2471
Description: A soundness justification theorem for df-eu 2473, 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 2472 for a proof that provides an example of how it can be achieved through the use of dvelim 2336. (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 1950 . . . . 5 (𝑦 = 𝑤 → (𝑥 = 𝑦𝑥 = 𝑤))
21bibi2d 332 . . . 4 (𝑦 = 𝑤 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑤)))
32albidv 1846 . . 3 (𝑦 = 𝑤 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑤)))
43cbvexv 2274 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑤𝑥(𝜑𝑥 = 𝑤))
5 equequ2 1950 . . . . 5 (𝑤 = 𝑧 → (𝑥 = 𝑤𝑥 = 𝑧))
65bibi2d 332 . . . 4 (𝑤 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ (𝜑𝑥 = 𝑧)))
76albidv 1846 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
87cbvexv 2274 . 2 (∃𝑤𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
94, 8bitri 264 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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