Theorem comraddd 8178

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 8172	. 2 |- ( ph -> ( B + C ) = ( C + B ) )
5	1, 4	eqtrd 2226	1 |- ( ph -> A = ( C + B ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164  (class class class)co 5919   CCcc 7872   + caddc 7877

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 7974

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

This theorem is referenced by:  mvrladdd  8388  hashfz  10895  bdtrilem  11385  clim2ser2  11484  fsumparts  11616  arisum  11644  divalglemnn  12062  phiprmpw  12363  mulgdir  13227  metrtri  14556  apdifflemr  15607