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Theorem breq 5150
Description: Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
Assertion
Ref Expression
breq (𝑅 = 𝑆 → (𝐴𝑅𝐵𝐴𝑆𝐵))

Proof of Theorem breq
StepHypRef Expression
1 eleq2 2828 . 2 (𝑅 = 𝑆 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2 df-br 5149 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3 df-br 5149 . 2 (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
41, 2, 33bitr4g 314 1 (𝑅 = 𝑆 → (𝐴𝑅𝐵𝐴𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814  df-br 5149
This theorem is referenced by:  breqi  5154  breqd  5159  poeq1  5600  soeq1  5618  freq1  5656  fveq1  6906  foeqcnvco  7320  f1eqcocnv  7321  isoeq2  7338  isoeq3  7339  eqfunresadj  7380  brfvopab  7490  ofreq  7701  supeq3  9487  oieq1  9550  ttrcleq  9747  dcomex  10485  axdc2lem  10486  brdom3  10566  brdom7disj  10569  brdom6disj  10570  dfrtrclrec2  15094  relexpindlem  15099  rtrclind  15101  shftfval  15106  isprs  18354  isdrs  18359  ispos  18372  istos  18476  efglem  19749  frgpuplem  19805  ordtval  23213  utop2nei  24275  utop3cls  24276  isucn2  24304  ucnima  24306  iducn  24308  ex-opab  30461  resspos  32941  acycgr0v  35133  prclisacycgr  35136  satf  35338  cureq  37583  poimirlem31  37638  poimir  37640  cosseq  38408  elrefrels3  38501  elcnvrefrels3  38517  elsymrels3  38536  elsymrels5  38538  eltrrels3  38562  eleqvrels3  38575  brabsb2  38844  rfovfvd  43992  fsovrfovd  43999  sprsymrelf  47420  sprsymrelfo  47422  upwlkbprop  47982
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