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

Theorem equvinv 2032
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2137, ax-13 2372. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)
Assertion
Ref Expression
equvinv (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinv
StepHypRef Expression
1 equequ1 2028 . . 3 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
21equsexvw 2008 . 2 (∃𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ 𝑥 = 𝑦)
32bicomi 223 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783
This theorem is referenced by:  ax8  2112  ax9  2120  ax13  2375  cossid  36598
  Copyright terms: Public domain W3C validator