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Metamath Proof Explorer

Theorem 3eqtr4a

Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

Ref Expression
Hypotheses 3eqtr4a.1 
|- A = B
3eqtr4a.2 
|- ( ph -> C = A )
3eqtr4a.3 
|- ( ph -> D = B )
Assertion 3eqtr4a 
|- ( ph -> C = D )

Proof

Step Hyp Ref Expression
1 3eqtr4a.1 
 |-  A = B
2 3eqtr4a.2 
 |-  ( ph -> C = A )
3 3eqtr4a.3 
 |-  ( ph -> D = B )
4 2 1 eqtrdi 
 |-  ( ph -> C = B )
5 4 3 eqtr4d 
 |-  ( ph -> C = D )