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

Theorem equvinvOLD 1964
 Description: Obsolete version of equvinv 1961 as of 11-Apr-2021. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2021, ax-13 2250. (Revised by Wolf Lammen, 10-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equvinvOLD (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinvOLD
StepHypRef Expression
1 equviniva 1962 . 2 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
2 equtrr 1951 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
32equcoms 1949 . . . 4 (𝑦 = 𝑧 → (𝑥 = 𝑧𝑥 = 𝑦))
43impcom 446 . . 3 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
54exlimiv 1860 . 2 (∃𝑧(𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
61, 5impbii 199 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∃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 1841  ax-6 1890  ax-7 1937 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator