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Metamath Proof Explorer

Theorem 3eqtrrd

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006) (Proof shortened by Andrew Salmon, 25-May-2011)

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

Proof

Step Hyp Ref Expression
1 3eqtrd.1 
 |-  ( ph -> A = B )
2 3eqtrd.2 
 |-  ( ph -> B = C )
3 3eqtrd.3 
 |-  ( ph -> C = D )
4 1 2 eqtrd 
 |-  ( ph -> A = C )
5 4 3 eqtr2d 
 |-  ( ph -> D = A )