Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqvreleqi Structured version   Visualization version   GIF version

Theorem eqvreleqi 37094
Description: Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
Hypothesis
Ref Expression
eqvreleqi.1 𝑅 = 𝑆
Assertion
Ref Expression
eqvreleqi ( EqvRel 𝑅 ↔ EqvRel 𝑆)

Proof of Theorem eqvreleqi
StepHypRef Expression
1 eqvreleqi.1 . 2 𝑅 = 𝑆
2 eqvreleq 37093 . 2 (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
31, 2ax-mp 5 1 ( EqvRel 𝑅 ↔ EqvRel 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542   EqvRel weqvrel 36680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-refrel 37003  df-symrel 37035  df-trrel 37065  df-eqvrel 37076
This theorem is referenced by:  dfcoeleqvrel  37113  eqvreldmqs2  37167  eldisjim2  37276  eqvrel0  37277  eqvrelid  37280
  Copyright terms: Public domain W3C validator