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Theorem breqan12rd 4600
 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 4599 . 2 ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
43ancoms 468 1 ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   class class class wbr 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584 This theorem is referenced by:  f1oweALT  7043  ledivdiv  10791  xltnegi  11921  ramub1lem1  15568  dvferm1  23552  dvferm2  23554  dvivthlem1  23575  ulmdvlem3  23960  gausslemma2dlem3  24893  lgsquad  24908  areacirclem4  32673  areacirclem5  32674  iccpartgt  39965
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