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

Theorem breqan12rd 5057
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 5056 . 2 ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
43ancoms 462 1 ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542   class class class wbr 5040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-un 3858  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041
This theorem is referenced by:  f1oweALT  7711  ledivdiv  11620  xltnegi  12705  ramub1lem1  16475  dvferm1  24750  dvferm2  24752  dvivthlem1  24773  ulmdvlem3  25162  gausslemma2dlem3  26117  lgsquad  26132  areacirclem4  35524  areacirclem5  35525  iccpartgt  44461
  Copyright terms: Public domain W3C validator