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Theorem breqdi 4859
Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
Hypotheses
Ref Expression
breq1d.1 (𝜑𝐴 = 𝐵)
breqdi.1 (𝜑𝐶𝐴𝐷)
Assertion
Ref Expression
breqdi (𝜑𝐶𝐵𝐷)

Proof of Theorem breqdi
StepHypRef Expression
1 breqdi.1 . 2 (𝜑𝐶𝐴𝐷)
2 breq1d.1 . . 3 (𝜑𝐴 = 𝐵)
32breqd 4855 . 2 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
41, 3mpbid 224 1 (𝜑𝐶𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653   class class class wbr 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2793  df-clel 2796  df-br 4845
This theorem is referenced by:  brfvimex  39101  brovmptimex  39102  ntrclsnvobr  39127  clsneibex  39177  neicvgbex  39187
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