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

Theorem equvini 2179
 Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require to be distinct from and . See equvin 1873 for a shorter proof requiring fewer axioms when is required to be distinct from and . (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
Assertion
Ref Expression
equvini

Proof of Theorem equvini
StepHypRef Expression
1 equtr 1865 . . . 4
2 equequ2 1868 . . . . . 6
32biimprd 227 . . . . 5
43anc2ri 561 . . . 4
51, 4syli 38 . . 3
6 19.8a 1935 . . 3
75, 6syl6 34 . 2
8 ax13 2141 . . 3
9 ax6e 2094 . . . . 5
109, 4eximii 1709 . . . 4
111019.35i 1741 . . 3
128, 11syl6 34 . 2
137, 12pm2.61i 168 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 371  wal 1442  wex 1663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091 This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668 This theorem is referenced by:  2ax6elem  2278
 Copyright terms: Public domain W3C validator