Theorem comraddd 7943

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 7937	. 2 |- ( ph -> ( B + C ) = ( C + B ) )
5	1, 4	eqtrd 2173	1 |- ( ph -> A = ( C + B ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481  (class class class)co 5782   CCcc 7642   + caddc 7647

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 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122  ax-addcom 7744

This theorem depends on definitions:  df-bi 116  df-cleq 2133

This theorem is referenced by:  mvrladdd  8153  hashfz  10599  bdtrilem  11042  clim2ser2  11139  fsumparts  11271  arisum  11299  divalglemnn  11651  phiprmpw  11934  metrtri  12585  apdifflemr  13415