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Theorem breqdi 5156
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 5152 . 2 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
41, 3mpbid 231 1 (𝜑𝐶𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   class class class wbr 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2718  df-clel 2804  df-br 5142
This theorem is referenced by:  rtrclreclem3  15011  episect  17739  dvef  25863  acopyeu  28589  isleagd  28603  0prjspn  41929  brfvimex  43334  brovmptimex  43335  ntrclsnvobr  43360  clsneibex  43410  neicvgbex  43420
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