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Theorem breqdi 5163
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 5159 . 2 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
41, 3mpbid 232 1 (𝜑𝐶𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537   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:  rtrclreclem3  15096  episect  17833  dvef  26033  acopyeu  28857  isleagd  28871  weiunso  36449  0prjspn  42615  brfvimex  44016  brovmptimex  44017  ntrclsnvobr  44042  clsneibex  44092  neicvgbex  44102
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