Metamath Proof Explorer
Description: Relationship between subtraction and addition. Based on subaddd .
(Contributed by Steven Nguyen, 8-Jan-2023)
|
|
Ref |
Expression |
|
Hypotheses |
resubaddd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
resubaddd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
|
resubaddd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
|
Assertion |
resubaddd |
⊢ ( 𝜑 → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
resubaddd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
resubaddd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
resubaddd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
4 |
|
resubadd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) ) |