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Theorem breqdi 4819
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 4815 . 2 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
41, 3mpbid 222 1 (𝜑𝐶𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632   class class class wbr 4804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2753  df-clel 2756  df-br 4805
This theorem is referenced by:  brfvimex  38826  brovmptimex  38827  ntrclsnvobr  38852  clsneibex  38902  neicvgbex  38912
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