Theorem addcomli 8434

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

Hypotheses

Ref	Expression
mul.1	|- A e. CC
mul.2	|- B e. CC
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 e. CC
2		mul.1	. . 3 |- A e. CC
3	1, 2	addcomi 8433	. 2 |- ( B + A ) = ( A + B )
4		addcomli.2	. 2 |- ( A + B ) = C
5	3, 4	eqtri 2255	1 |- ( B + A ) = C

Syntax hints:   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   + caddc 8146

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216  ax-addcom 8243

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  negsubdi2i  8575  1p2e3  9389  peano2z  9630  4t4e16  9825  6t3e18  9831  6t5e30  9833  7t3e21  9836  7t4e28  9837  7t6e42  9839  7t7e49  9840  8t3e24  9842  8t4e32  9843  8t5e40  9844  8t8e64  9847  9t3e27  9849  9t4e36  9850  9t5e45  9851  9t6e54  9852  9t7e63  9853  9t8e72  9854  9t9e81  9855  4bc3eq4  11161  n2dvdsm1  12624  bitsfzo  12666  6gcd4e2  12716  gcdi  13143  2exp8  13158  2exp16  13160  eulerid  15793  cosq23lt0  15824  binom4  15970  lgsdir2lem1  16027  m1lgs  16084  2lgsoddprmlem3d  16109  ex-exp  16621  ex-bc  16623  ex-gcd  16625