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Theorem breqdi 5054
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 5050 . 2 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
41, 3mpbid 235 1 (𝜑𝐶𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538   class class class wbr 5039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2814  df-clel 2892  df-br 5040
This theorem is referenced by:  rtrclreclem3  14398  episect  17034  dvef  24562  acopyeu  26607  isleagd  26621  0prjspn  39421  brfvimex  40539  brovmptimex  40540  ntrclsnvobr  40565  clsneibex  40615  neicvgbex  40625
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