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Metamath Proof Explorer

Theorem eqneltrrd

Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017) (Proof shortened by Wolf Lammen, 13-Nov-2019)

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

Proof

Step Hyp Ref Expression
1 eqneltrrd.1 
 |-  ( ph -> A = B )
2 eqneltrrd.2 
 |-  ( ph -> -. A e. C )
3 1 eqcomd 
 |-  ( ph -> B = A )
4 3 2 eqneltrd 
 |-  ( ph -> -. B e. C )