Theorem addcomli 8102

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 8101	. 2 |- ( B + A ) = ( A + B )
4		addcomli.2	. 2 |- ( A + B ) = C
5	3, 4	eqtri 2198	1 |- ( B + A ) = C

Syntax hints:   = wceq 1353   e. wcel 2148  (class class class)co 5875   CCcc 7809   + caddc 7814

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159  ax-addcom 7911

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  negsubdi2i  8243  1p2e3  9053  peano2z  9289  4t4e16  9482  6t3e18  9488  6t5e30  9490  7t3e21  9493  7t4e28  9494  7t6e42  9496  7t7e49  9497  8t3e24  9499  8t4e32  9500  8t5e40  9501  8t8e64  9504  9t3e27  9506  9t4e36  9507  9t5e45  9508  9t6e54  9509  9t7e63  9510  9t8e72  9511  9t9e81  9512  4bc3eq4  10753  n2dvdsm1  11918  6gcd4e2  11996  eulerid  14226  cosq23lt0  14257  binom4  14400  lgsdir2lem1  14432  m1lgs  14455  2lgsoddprmlem3d  14461  ex-exp  14482  ex-bc  14484  ex-gcd  14486