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Theorem eujust 2649
Description: Soundness justification theorem for eu6 2652 when this was the definition of the unique existential quantifier (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). See eujustALT 2650 for a proof that provides an example of how it can be achieved through the use of dvelim 2465. (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 2024 . . . . 5 (𝑦 = 𝑤 → (𝑥 = 𝑦𝑥 = 𝑤))
21bibi2d 344 . . . 4 (𝑦 = 𝑤 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑤)))
32albidv 1912 . . 3 (𝑦 = 𝑤 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑤)))
43cbvexvw 2035 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑤𝑥(𝜑𝑥 = 𝑤))
5 equequ2 2024 . . . . 5 (𝑤 = 𝑧 → (𝑥 = 𝑤𝑥 = 𝑧))
65bibi2d 344 . . . 4 (𝑤 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ (𝜑𝑥 = 𝑧)))
76albidv 1912 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
87cbvexvw 2035 . 2 (∃𝑤𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
94, 8bitri 276 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by: (None)
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