Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eujust Structured version   Visualization version   GIF version

Theorem eujust 2619
 Description: A soundness justification theorem for df-eu 2621, 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 2620 for a proof that provides an example of how it can be achieved through the use of dvelim 2486. (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 2110 . . . . 5 (𝑦 = 𝑤 → (𝑥 = 𝑦𝑥 = 𝑤))
21bibi2d 331 . . . 4 (𝑦 = 𝑤 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑤)))
32albidv 2000 . . 3 (𝑦 = 𝑤 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑤)))
43cbvexv 2434 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑤𝑥(𝜑𝑥 = 𝑤))
5 equequ2 2110 . . . . 5 (𝑤 = 𝑧 → (𝑥 = 𝑤𝑥 = 𝑧))
65bibi2d 331 . . . 4 (𝑤 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ (𝜑𝑥 = 𝑧)))
76albidv 2000 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
87cbvexv 2434 . 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 1628  ∃wex 1851 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-11 2189  ax-12 2202  ax-13 2407 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator