Description: A transitive law for class equality. (Contributed by NM, 20-May-2005) (Proof shortened by Andrew Salmon, 25-May-2011)
Ref | Expression | ||
---|---|---|---|
Assertion | eqtr2 | ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom | ⊢ ( 𝐴 = 𝐵 ↔ 𝐵 = 𝐴 ) | |
2 | eqtr | ⊢ ( ( 𝐵 = 𝐴 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 ) | |
3 | 1 2 | sylanb | ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 ) |