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Theorem comraddd 8109
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
StepHypRef 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 )
42, 3addcomd 8103 . 2  |-  ( ph  ->  ( B  +  C
)  =  ( C  +  B ) )
51, 4eqtrd 2210 1  |-  ( ph  ->  A  =  ( C  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148  (class class class)co 5871   CCcc 7805    + caddc 7810
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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159  ax-addcom 7907
This theorem depends on definitions:  df-bi 117  df-cleq 2170
This theorem is referenced by:  mvrladdd  8319  hashfz  10793  bdtrilem  11239  clim2ser2  11338  fsumparts  11470  arisum  11498  divalglemnn  11914  phiprmpw  12213  mulgdir  12945  metrtri  13739  apdifflemr  14646
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