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Theorem breqdi 4598
 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 4594 . 2 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
41, 3mpbid 221 1 (𝜑𝐶𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = 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-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-br 4584 This theorem is referenced by:  brfvimex  37344  brovmptimex  37345  ntrclsnvobr  37370  clsneibex  37420  neicvgbex  37430
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