Description: Move LHS left addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mvlraddd.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
mvlraddd.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | ||
mvlraddd.3 | ⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = 𝐶 ) | ||
Assertion | mvlladdd | ⊢ ( 𝜑 → 𝐵 = ( 𝐶 − 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvlraddd.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
2 | mvlraddd.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | |
3 | mvlraddd.3 | ⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = 𝐶 ) | |
4 | 2 1 | pncand | ⊢ ( 𝜑 → ( ( 𝐵 + 𝐴 ) − 𝐴 ) = 𝐵 ) |
5 | 1 2 | addcomd | ⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) ) |
6 | 5 3 | eqtr3d | ⊢ ( 𝜑 → ( 𝐵 + 𝐴 ) = 𝐶 ) |
7 | 6 | oveq1d | ⊢ ( 𝜑 → ( ( 𝐵 + 𝐴 ) − 𝐴 ) = ( 𝐶 − 𝐴 ) ) |
8 | 4 7 | eqtr3d | ⊢ ( 𝜑 → 𝐵 = ( 𝐶 − 𝐴 ) ) |