ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elequ2 GIF version

Theorem elequ2 1617
Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
elequ2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Proof of Theorem elequ2
StepHypRef Expression
1 ax-14 1421 . 2 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
2 ax-14 1421 . . 3 (𝑦 = 𝑥 → (𝑧𝑦𝑧𝑥))
32equcoms 1610 . 2 (𝑥 = 𝑦 → (𝑧𝑦𝑧𝑥))
41, 3impbid 124 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ie2 1399  ax-8 1411  ax-14 1421  ax-17 1435  ax-i9 1439
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  elsb4  1869  dveel2  1913  axext3  2039  axext4  2040  bm1.1  2041  bm1.3ii  3906  nalset  3915  zfun  4199  fv3  5225  tfrlemisucaccv  5970  bdsepnft  10394  bdsepnfALT  10396  bdbm1.3ii  10398  bj-nalset  10402  bj-nnelirr  10465  strcollnft  10496  strcollnfALT  10498
  Copyright terms: Public domain W3C validator