Theorem addcomli 8164

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 8163	. 2 ⊢ (𝐵 + 𝐴) = (𝐴 + 𝐵)
4		addcomli.2	. 2 ⊢ (𝐴 + 𝐵) = 𝐶
5	3, 4	eqtri 2214	1 ⊢ (𝐵 + 𝐴) = 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2164  (class class class)co 5918  ℂcc 7870   + caddc 7875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175  ax-addcom 7972

This theorem depends on definitions:  df-bi 117  df-cleq 2186

This theorem is referenced by:  negsubdi2i  8305  1p2e3  9116  peano2z  9353  4t4e16  9546  6t3e18  9552  6t5e30  9554  7t3e21  9557  7t4e28  9558  7t6e42  9560  7t7e49  9561  8t3e24  9563  8t4e32  9564  8t5e40  9565  8t8e64  9568  9t3e27  9570  9t4e36  9571  9t5e45  9572  9t6e54  9573  9t7e63  9574  9t8e72  9575  9t9e81  9576  4bc3eq4  10844  n2dvdsm1  12054  6gcd4e2  12132  eulerid  14937  cosq23lt0  14968  binom4  15111  lgsdir2lem1  15144  m1lgs  15192  2lgsoddprmlem3d  15198  ex-exp  15219  ex-bc  15221  ex-gcd  15223