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Metamath Proof Explorer

Theorem 3eqtr3d

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

Ref Expression
Hypotheses 3eqtr3d.1 ⊢ φ → A = B
3eqtr3d.2 ⊢ φ → A = C
3eqtr3d.3 ⊢ φ → B = D
Assertion 3eqtr3d ⊢ φ → C = D

Proof

Step Hyp Ref Expression
1 3eqtr3d.1 ⊢ φ → A = B
2 3eqtr3d.2 ⊢ φ → A = C
3 3eqtr3d.3 ⊢ φ → B = D
4 1 2 eqtr3d ⊢ φ → B = C
5 4 3 eqtr3d ⊢ φ → C = D