ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  breqd GIF version

Theorem breqd 3775
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 3766 . 2 (𝐴 = 𝐵 → (𝐶𝐴𝐷𝐶𝐵𝐷))
31, 2syl 14 1 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243   class class class wbr 3764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-br 3765
This theorem is referenced by:  breq123d  3778  sbcbr12g  3814  sprmpt2  5857  shftfibg  9421  shftfib  9424  2shfti  9432
  Copyright terms: Public domain W3C validator