This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 3eqtrd

Description: A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995)

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

Proof

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