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Axiom ax-addass 7394
Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7354. Proofs should normally use addass 7419 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 7295 . . . 4 class
31, 2wcel 1436 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1436 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1436 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 922 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 caddc 7300 . . . . 5 class +
101, 4, 9co 5615 . . . 4 class (𝐴 + 𝐵)
1110, 6, 9co 5615 . . 3 class ((𝐴 + 𝐵) + 𝐶)
124, 6, 9co 5615 . . . 4 class (𝐵 + 𝐶)
131, 12, 9co 5615 . . 3 class (𝐴 + (𝐵 + 𝐶))
1411, 13wceq 1287 . 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  7419
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