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Theorem breqd 3940
Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
Hypothesis
Ref Expression
breq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
breqd (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))

Proof of Theorem breqd
StepHypRef Expression
1 breq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 breq 3931 . 2 (𝐴 = 𝐵 → (𝐶𝐴𝐷𝐶𝐵𝐷))
31, 2syl 14 1 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = 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:  breq123d  3943  breqdi  3944  sbcbr12g  3983  supeq123d  6878  shftfibg  10604  shftfib  10607  2shfti  10615  lmbr  12396
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