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Metamath Proof Explorer

Theorem add42d

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 ) ) )