MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  breqan12rd Structured version   Visualization version   GIF version

Theorem breqan12rd 5074
Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
Hypotheses
Ref Expression
breq1d.1 (𝜑𝐴 = 𝐵)
breqan12i.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
breqan12rd ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Proof of Theorem breqan12rd
StepHypRef Expression
1 breq1d.1 . . 3 (𝜑𝐴 = 𝐵)
2 breqan12i.2 . . 3 (𝜓𝐶 = 𝐷)
31, 2breqan12d 5073 . 2 ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
43ancoms 459 1 ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528   class class class wbr 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058
This theorem is referenced by:  f1oweALT  7662  ledivdiv  11517  xltnegi  12597  ramub1lem1  16350  dvferm1  24509  dvferm2  24511  dvivthlem1  24532  ulmdvlem3  24917  gausslemma2dlem3  25871  lgsquad  25886  areacirclem4  34866  areacirclem5  34867  iccpartgt  43464
  Copyright terms: Public domain W3C validator