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Metamath Proof Explorer

Theorem 3eqtr2d

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006)

Ref Expression
Hypotheses 3eqtr2d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
3eqtr2d.2 ⊢ ( 𝜑 → 𝐶 = 𝐵 )
3eqtr2d.3 ⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion 3eqtr2d ( 𝜑 → 𝐴 = 𝐷 )

Proof

Step Hyp Ref Expression
1 3eqtr2d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 3eqtr2d.2 ⊢ ( 𝜑 → 𝐶 = 𝐵 )
3 3eqtr2d.3 ⊢ ( 𝜑 → 𝐶 = 𝐷 )
4 1 2 eqtr4d ⊢ ( 𝜑 → 𝐴 = 𝐶 )
5 4 3 eqtrd ⊢ ( 𝜑 → 𝐴 = 𝐷 )