Metamath Proof Explorer
Description: Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
addd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
addd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
addd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
|
|
add4d.4 |
⊢ ( 𝜑 → 𝐷 ∈ ℂ ) |
|
Assertion |
add42d |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
addd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
addd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
addd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
4 |
|
add4d.4 |
⊢ ( 𝜑 → 𝐷 ∈ ℂ ) |
5 |
|
add42 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) ) |
6 |
1 2 3 4 5
|
syl22anc |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) ) |