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

Theorem equviniva 2113
 Description: A modified version of the forward implication of equvinv 2112 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.)
Assertion
Ref Expression
equviniva (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equviniva
StepHypRef Expression
1 ax6evr 2097 . 2 𝑧 𝑦 = 𝑧
2 equtr 2103 . . . 4 (𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))
32ancrd 578 . . 3 (𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)))
43eximdv 1995 . 2 (𝑥 = 𝑦 → (∃𝑧 𝑦 = 𝑧 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧)))
51, 4mpi 20 1 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1853 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854 This theorem is referenced by:  ax13lem1  2393  nfeqf  2446  bj-ssbequ2  32949  wl-ax13lem1  33600
 Copyright terms: Public domain W3C validator