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Theorem subadd 8078
Description: Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
Assertion
Ref Expression
subadd ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))

Proof of Theorem subadd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 subval 8067 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
21eqeq1d 2166 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
323adant3 1002 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
4 negeu 8066 . . . . 5 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
5 oveq2 5832 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 + 𝑥) = (𝐵 + 𝐶))
65eqeq1d 2166 . . . . . 6 (𝑥 = 𝐶 → ((𝐵 + 𝑥) = 𝐴 ↔ (𝐵 + 𝐶) = 𝐴))
76riota2 5802 . . . . 5 ((𝐶 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
84, 7sylan2 284 . . . 4 ((𝐶 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
983impb 1181 . . 3 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
1093com13 1190 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
113, 10bitr4d 190 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963   = wceq 1335  wcel 2128  ∃!wreu 2437  crio 5779  (class class class)co 5824  cc 7730   + caddc 7735  cmin 8046
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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4496  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-sub 8048
This theorem is referenced by:  subadd2  8079  subsub23  8080  pncan  8081  pncan3  8083  addsubeq4  8090  subsub2  8103  renegcl  8136  subaddi  8162  subaddd  8204  fzen  9945  nn0ennn  10332  cos2t  11647  cos2tsin  11648  odd2np1  11763  divalgb  11815  sincosq1eq  13160  coskpi  13169
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