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Metamath Proof Explorer

Theorem 3eqtr4d

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995) (Proof shortened by Andrew Salmon, 25-May-2011)

Ref Expression
Hypotheses 3eqtr4d.1 
|- ( ph -> A = B )
3eqtr4d.2 
|- ( ph -> C = A )
3eqtr4d.3 
|- ( ph -> D = B )
Assertion 3eqtr4d 
|- ( ph -> C = D )

Proof

Step Hyp Ref Expression
1 3eqtr4d.1 
 |-  ( ph -> A = B )
2 3eqtr4d.2 
 |-  ( ph -> C = A )
3 3eqtr4d.3 
 |-  ( ph -> D = B )
4 3 1 eqtr4d 
 |-  ( ph -> D = A )
5 2 4 eqtr4d 
 |-  ( ph -> C = D )