Theorem addcomli 7253

Description: Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)

Hypotheses

Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ
mul.2	⊢ 𝐵 ∈ ℂ
addcomli.2	⊢ (𝐴 + 𝐵) = 𝐶

Assertion

Ref	Expression
addcomli	⊢ (𝐵 + 𝐴) = 𝐶

Proof of Theorem addcomli

Step	Hyp	Ref	Expression
1		mul.2	. . 3 ⊢ 𝐵 ∈ ℂ
2		mul.1	. . 3 ⊢ 𝐴 ∈ ℂ
3	1, 2	addcomi 7252	. 2 ⊢ (𝐵 + 𝐴) = (𝐴 + 𝐵)
4		addcomli.2	. 2 ⊢ (𝐴 + 𝐵) = 𝐶
5	3, 4	eqtri 2101	1 ⊢ (𝐵 + 𝐴) = 𝐶

Syntax hints:   = wceq 1284   ∈ wcel 1433  (class class class)co 5532  ℂcc 6979   + caddc 6984

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063  ax-addcom 7076

This theorem depends on definitions:  df-bi 115  df-cleq 2074

This theorem is referenced by:  negsubdi2i  7394  1p2e3  8166  peano2z  8387  4t4e16  8575  6t3e18  8581  6t5e30  8583  7t3e21  8586  7t4e28  8587  7t6e42  8589  7t7e49  8590  8t3e24  8592  8t4e32  8593  8t5e40  8594  8t8e64  8597  9t3e27  8599  9t4e36  8600  9t5e45  8601  9t6e54  8602  9t7e63  8603  9t8e72  8604  9t9e81  8605  4bc3eq4  9700  n2dvdsm1  10313  6gcd4e2  10384  ex-bc  10566  ex-gcd  10568