Theorem comraddd 8483

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 8477	. 2 |- ( ph -> ( B + C ) = ( C + B ) )
5	1, 4	eqtrd 2271	1 |- ( ph -> A = ( C + B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   + caddc 8182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220  ax-addcom 8279

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  mvrladdd  8693  hashfz  11262  bdtrilem  12005  clim2ser2  12104  fsumparts  12237  arisum  12265  divalglemnn  12685  phiprmpw  13000  mulgdir  13957  metrtri  15478  apdifflemr  17096