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Metamath Proof Explorer

Theorem 3eqtr3rd

Description: A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006)

Ref Expression
Hypotheses 3eqtr3d.1 
|- ( ph -> A = B )
3eqtr3d.2 
|- ( ph -> A = C )
3eqtr3d.3 
|- ( ph -> B = D )
Assertion 3eqtr3rd 
|- ( ph -> D = C )

Proof

Step Hyp Ref Expression
1 3eqtr3d.1 
 |-  ( ph -> A = B )
2 3eqtr3d.2 
 |-  ( ph -> A = C )
3 3eqtr3d.3 
 |-  ( ph -> B = D )
4 1 2 eqtr3d 
 |-  ( ph -> B = C )
5 3 4 eqtr3d 
 |-  ( ph -> D = C )