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Axiom ax-addass 6984
Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 6944. Proofs should normally use addass 7009 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-addass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Detailed syntax breakdown of Axiom ax-addass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 6885 . . . 4 class
31, 2wcel 1393 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1393 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1393 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 885 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 caddc 6890 . . . . 5 class +
101, 4, 9co 5512 . . . 4 class (𝐴 + 𝐵)
1110, 6, 9co 5512 . . 3 class ((𝐴 + 𝐵) + 𝐶)
124, 6, 9co 5512 . . . 4 class (𝐵 + 𝐶)
131, 12, 9co 5512 . . 3 class (𝐴 + (𝐵 + 𝐶))
1411, 13wceq 1243 . 2 wff ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff set class
This axiom is referenced by:  addass  7009
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