Metamath Proof Explorer


Theorem eqnbrtrd

Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021)

Ref Expression
Hypotheses eqnbrtrd.1
|- ( ph -> A = B )
eqnbrtrd.2
|- ( ph -> -. B R C )
Assertion eqnbrtrd
|- ( ph -> -. A R C )

Proof

Step Hyp Ref Expression
1 eqnbrtrd.1
 |-  ( ph -> A = B )
2 eqnbrtrd.2
 |-  ( ph -> -. B R C )
3 1 breq1d
 |-  ( ph -> ( A R C <-> B R C ) )
4 2 3 mtbird
 |-  ( ph -> -. A R C )