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

Theorem breqdi 4015
Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
Hypotheses
Ref Expression
breq1d.1  |-  ( ph  ->  A  =  B )
breqdi.1  |-  ( ph  ->  C A D )
Assertion
Ref Expression
breqdi  |-  ( ph  ->  C B D )

Proof of Theorem breqdi
StepHypRef Expression
1 breqdi.1 . 2  |-  ( ph  ->  C A D )
2 breq1d.1 . . 3  |-  ( ph  ->  A  =  B )
32breqd 4011 . 2  |-  ( ph  ->  ( C A D  <-> 
C B D ) )
41, 3mpbid 147 1  |-  ( ph  ->  C B D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   class class class wbr 4000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-br 4001
This theorem is referenced by:  dvef  13821
  Copyright terms: Public domain W3C validator