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Theorem breqdi 3944
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 3940 . 2 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
41, 3mpbid 146 1 (𝜑𝐶𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331   class class class wbr 3929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-br 3930
This theorem is referenced by:  dvef  12871
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