Theorem comraddd 8231

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 8225	. 2 |- ( ph -> ( B + C ) = ( C + B ) )
5	1, 4	eqtrd 2238	1 |- ( ph -> A = ( C + B ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176  (class class class)co 5946   CCcc 7925   + caddc 7930

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 1470  ax-gen 1472  ax-4 1533  ax-17 1549  ax-ext 2187  ax-addcom 8027

This theorem depends on definitions:  df-bi 117  df-cleq 2198

This theorem is referenced by:  mvrladdd  8441  hashfz  10968  bdtrilem  11583  clim2ser2  11682  fsumparts  11814  arisum  11842  divalglemnn  12262  phiprmpw  12577  mulgdir  13523  metrtri  14882  apdifflemr  16023