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Theorem pncan3oi 10148
Description: Subtraction and addition of equals. Almost but not exactly the same as pncan3i 10209 and pncan 10138, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 10244. (Contributed by David A. Wheeler, 11-Oct-2018.)
Hypotheses
Ref Expression
pncan3oi.1 𝐴 ∈ ℂ
pncan3oi.2 𝐵 ∈ ℂ
Assertion
Ref Expression
pncan3oi ((𝐴 + 𝐵) − 𝐵) = 𝐴

Proof of Theorem pncan3oi
StepHypRef Expression
1 pncan3oi.1 . 2 𝐴 ∈ ℂ
2 pncan3oi.2 . 2 𝐵 ∈ ℂ
3 pncan 10138 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴)
41, 2, 3mp2an 703 1 ((𝐴 + 𝐵) − 𝐵) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1976  (class class class)co 6527  cc 9790   + caddc 9795  cmin 10117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-po 4949  df-so 4950  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-ltxr 9935  df-sub 10119
This theorem is referenced by:  mvrraddi  10149  mvlladdi  10150  climcndslem1  14366  3dvds  14836  3dvdsOLD  14837  1259prm  15627  2503prm  15631  ovolicc2lem4  23012  eff1o  24016  log2tlbnd  24389  birthday  24398  basellem8  24531  ppiublem2  24645  ppiub  24646  chtub  24654  bposlem6  24731  bposlem8  24733  ex-ind-dvds  26476  lnfn0i  28091  lmatfvlem  29015  quad3  30624  poimirlem16  32391  poimirlem17  32392  poimirlem19  32394  poimirlem20  32395  fdc  32507  heiborlem6  32581  areaquad  36617  inductionexd  37269  stoweidlem34  38724  fouriersw  38921  mvlraddi  42279  mvrladdi  42281
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