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Metamath Proof Explorer

Theorem eqnetrrd

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012)

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

Proof

Step Hyp Ref Expression
1 eqnetrrd.1 
 |-  ( ph -> A = B )
2 eqnetrrd.2 
 |-  ( ph -> A =/= C )
3 1 eqcomd 
 |-  ( ph -> B = A )
4 3 2 eqnetrd 
 |-  ( ph -> B =/= C )