ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  comraddd GIF version

Theorem comraddd 8032
Description: Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
Hypotheses
Ref Expression
comraddd.1 (𝜑𝐵 ∈ ℂ)
comraddd.2 (𝜑𝐶 ∈ ℂ)
comraddd.3 (𝜑𝐴 = (𝐵 + 𝐶))
Assertion
Ref Expression
comraddd (𝜑𝐴 = (𝐶 + 𝐵))

Proof of Theorem comraddd
StepHypRef Expression
1 comraddd.3 . 2 (𝜑𝐴 = (𝐵 + 𝐶))
2 comraddd.1 . . 3 (𝜑𝐵 ∈ ℂ)
3 comraddd.2 . . 3 (𝜑𝐶 ∈ ℂ)
42, 3addcomd 8026 . 2 (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐵))
51, 4eqtrd 2190 1 (𝜑𝐴 = (𝐶 + 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wcel 2128  (class class class)co 5824  cc 7730   + caddc 7735
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 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139  ax-addcom 7832
This theorem depends on definitions:  df-bi 116  df-cleq 2150
This theorem is referenced by:  mvrladdd  8242  hashfz  10695  bdtrilem  11138  clim2ser2  11235  fsumparts  11367  arisum  11395  divalglemnn  11809  phiprmpw  12097  metrtri  12788  apdifflemr  13629
  Copyright terms: Public domain W3C validator