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 )