ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eleq1a GIF version

Theorem eleq1a 2125
Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
Assertion
Ref Expression
eleq1a (𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))

Proof of Theorem eleq1a
StepHypRef Expression
1 eleq1 2116 . 2 (𝐶 = 𝐴 → (𝐶𝐵𝐴𝐵))
21biimprcd 153 1 (𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409
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-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  elex22  2586  elex2  2587  reu6  2752  disjne  3300  ssimaex  5261  fnex  5410  f1ocnv2d  5731  tfrlem8  5964  eroprf  6229  ac6sfi  6382  recclnq  6547  prnmaddl  6645  renegcl  7334  nn0ind-raph  8413  iccid  8894  bj-nn0suc  10448  bj-inf2vnlem2  10455  bj-nn0sucALT  10462
  Copyright terms: Public domain W3C validator