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Axiom ax-addass 7690
Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7648. Proofs should normally use addass 7718 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 7586 . . . 4 class
31, 2wcel 1465 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1465 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1465 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 947 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 caddc 7591 . . . . 5 class +
101, 4, 9co 5742 . . . 4 class (𝐴 + 𝐵)
1110, 6, 9co 5742 . . 3 class ((𝐴 + 𝐵) + 𝐶)
124, 6, 9co 5742 . . . 4 class (𝐵 + 𝐶)
131, 12, 9co 5742 . . 3 class (𝐴 + (𝐵 + 𝐶))
1411, 13wceq 1316 . 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  7718
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