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Metamath Proof Explorer

Theorem 3eqtr4rd

Description: A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995)

Ref Expression
Hypotheses 3eqtr4d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
3eqtr4d.2 ⊢ ( 𝜑 → 𝐶 = 𝐴 )
3eqtr4d.3 ⊢ ( 𝜑 → 𝐷 = 𝐵 )
Assertion 3eqtr4rd ( 𝜑 → 𝐷 = 𝐶 )

Proof

Step Hyp Ref Expression
1 3eqtr4d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 3eqtr4d.2 ⊢ ( 𝜑 → 𝐶 = 𝐴 )
3 3eqtr4d.3 ⊢ ( 𝜑 → 𝐷 = 𝐵 )
4 3 1 eqtr4d ⊢ ( 𝜑 → 𝐷 = 𝐴 )
5 4 2 eqtr4d ⊢ ( 𝜑 → 𝐷 = 𝐶 )