Theorem comraddd 8314

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 8308	. 2 |- ( ph -> ( B + C ) = ( C + B ) )
5	1, 4	eqtrd 2262	1 |- ( ph -> A = ( C + B ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200  (class class class)co 6007   CCcc 8008   + caddc 8013

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211  ax-addcom 8110

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  mvrladdd  8524  hashfz  11056  bdtrilem  11766  clim2ser2  11865  fsumparts  11997  arisum  12025  divalglemnn  12445  phiprmpw  12760  mulgdir  13707  metrtri  15067  apdifflemr  16503