Theorem addcomli 8461

Description: Addition is commutative. (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 8460	. 2 |- ( B + A ) = ( A + B )
4		addcomli.2	. 2 |- ( A + B ) = C
5	3, 4	eqtri 2259	1 |- ( B + A ) = C

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   + caddc 8172

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 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220  ax-addcom 8269

This theorem depends on definitions:  df-bi 117  df-cleq 2231

This theorem is referenced by:  negsubdi2i  8602  1p2e3  9418  peano2z  9659  4t4e16  9854  6t3e18  9860  6t5e30  9862  7t3e21  9865  7t4e28  9866  7t6e42  9868  7t7e49  9869  8t3e24  9871  8t4e32  9872  8t5e40  9873  8t8e64  9876  9t3e27  9878  9t4e36  9879  9t5e45  9880  9t6e54  9881  9t7e63  9882  9t8e72  9883  9t9e81  9884  4bc3eq4  11190  n2dvdsm1  12658  bitsfzo  12700  6gcd4e2  12750  gcdi  13177  2exp8  13192  2exp16  13194  eulerid  15826  cosq23lt0  15857  binom4  16004  lgsdir2lem1  16061  m1lgs  16118  2lgsoddprmlem3d  16143  ex-exp  16655  ex-bc  16657  ex-gcd  16659