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

Theorem breqd 3804
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 3795 . 2 (𝐴 = 𝐵 → (𝐶𝐴𝐷𝐶𝐵𝐷))
31, 2syl 14 1 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285   class class class wbr 3793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-br 3794
This theorem is referenced by:  breq123d  3807  sbcbr12g  3843  sprmpt2  5891  supeq123d  6463  shftfibg  9846  shftfib  9849  2shfti  9857
  Copyright terms: Public domain W3C validator