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

Theorem releqi 5125
 Description: Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
Hypothesis
Ref Expression
releqi.1 𝐴 = 𝐵
Assertion
Ref Expression
releqi (Rel 𝐴 ↔ Rel 𝐵)

Proof of Theorem releqi
StepHypRef Expression
1 releqi.1 . 2 𝐴 = 𝐵
2 releq 5124 . 2 (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
31, 2ax-mp 5 1 (Rel 𝐴 ↔ Rel 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-rel 5045 This theorem is referenced by:  reliun  5162  reluni  5164  relint  5165  reldmmpt2  6669  wfrrel  7307  tfrlem6  7365  relsdom  7848  cda1dif  8881  0rest  15913  firest  15916  2oppchomf  16207  oppchofcl  16723  oyoncl  16733  releqg  17464  reldvdsr  18467  restbas  20772  hlimcaui  27477  relbigcup  31174  fnsingle  31196  funimage  31205  colinrel  31334  neicvgnvor  37434
 Copyright terms: Public domain W3C validator