Metamath Proof Explorer
Description: A subtraction law: Exchanging the subtrahend and the result of the
subtraction. (Contributed by BJ, 6-Jun-2019)
|
|
Ref |
Expression |
|
Hypotheses |
addlsub.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
addlsub.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
addlsub.c |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
|
Assertion |
subexsub |
⊢ ( 𝜑 → ( 𝐴 = ( 𝐶 − 𝐵 ) ↔ 𝐵 = ( 𝐶 − 𝐴 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
addlsub.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
addlsub.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
addlsub.c |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
4 |
1 2 3
|
addlsub |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) = 𝐶 ↔ 𝐴 = ( 𝐶 − 𝐵 ) ) ) |
5 |
1 2 3
|
addrsub |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) = 𝐶 ↔ 𝐵 = ( 𝐶 − 𝐴 ) ) ) |
6 |
4 5
|
bitr3d |
⊢ ( 𝜑 → ( 𝐴 = ( 𝐶 − 𝐵 ) ↔ 𝐵 = ( 𝐶 − 𝐴 ) ) ) |