Axiom ax-addcom 7849

Description: Addition commutes. Axiom for real and complex numbers, justified by Theorem axaddcom 7807. Proofs should normally use addcom 8031 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)

Assertion

Ref	Expression
ax-addcom	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Detailed syntax breakdown of Axiom ax-addcom

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7747	. . . 4 class ℂ
3	1, 2	wcel 2136	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2136	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 103	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		caddc 7752	. . . 4 class +
8	1, 4, 7	co 5841	. . 3 class (𝐴 + 𝐵)
9	4, 1, 7	co 5841	. . 3 class (𝐵 + 𝐴)
10	8, 9	wceq 1343	. 2 wff (𝐴 + 𝐵) = (𝐵 + 𝐴)
11	6, 10	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

This axiom is referenced by:  addcom  8031