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Theorem breqan12rd 5091
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 5090 . 2 ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
43ancoms 459 1 ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539   class class class wbr 5074
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075
This theorem is referenced by:  f1oweALT  7815  ledivdiv  11864  xltnegi  12950  ramub1lem1  16727  dvferm1  25149  dvferm2  25151  dvivthlem1  25172  ulmdvlem3  25561  gausslemma2dlem3  26516  lgsquad  26531  areacirclem4  35868  areacirclem5  35869  iccpartgt  44879
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