Theorem comraddd 8055

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 8049	. 2 |- ( ph -> ( B + C ) = ( C + B ) )
5	1, 4	eqtrd 2198	1 |- ( ph -> A = ( C + B ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   + caddc 7756

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 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147  ax-addcom 7853

This theorem depends on definitions:  df-bi 116  df-cleq 2158

This theorem is referenced by:  mvrladdd  8265  hashfz  10734  bdtrilem  11180  clim2ser2  11279  fsumparts  11411  arisum  11439  divalglemnn  11855  phiprmpw  12154  metrtri  13017  apdifflemr  13926