Theorem comraddd 7912

Description: Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)

Hypotheses

Ref	Expression
comraddd.1	|- ( ph -> B e. CC )
comraddd.2	|- ( ph -> C e. CC )
comraddd.3	|- ( ph -> A = ( B + C ) )

Assertion

Ref	Expression
comraddd	|- ( ph -> A = ( C + B ) )

Proof of Theorem comraddd

Step	Hyp	Ref	Expression
1		comraddd.3	. 2 |- ( ph -> A = ( B + C ) )
2		comraddd.1	. . 3 |- ( ph -> B e. CC )
3		comraddd.2	. . 3 |- ( ph -> C e. CC )
4	2, 3	addcomd 7906	. 2 |- ( ph -> ( B + C ) = ( C + B ) )
5	1, 4	eqtrd 2170	1 |- ( ph -> A = ( C + B ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480  (class class class)co 5767   CCcc 7611   + caddc 7616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2119  ax-addcom 7713

This theorem depends on definitions:  df-bi 116  df-cleq 2130

This theorem is referenced by:  mvrladdd  8122  hashfz  10560  bdtrilem  11003  clim2ser2  11100  fsumparts  11232  arisum  11260  divalglemnn  11604  phiprmpw  11887  metrtri  12535