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Theorem subadd 7413
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 7402 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
21eqeq1d 2091 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
323adant3 959 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
4 negeu 7401 . . . . 5 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
5 oveq2 5571 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 + 𝑥) = (𝐵 + 𝐶))
65eqeq1d 2091 . . . . . 6 (𝑥 = 𝐶 → ((𝐵 + 𝑥) = 𝐴 ↔ (𝐵 + 𝐶) = 𝐴))
76riota2 5541 . . . . 5 ((𝐶 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
84, 7sylan2 280 . . . 4 ((𝐶 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
983impb 1135 . . 3 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
1093com13 1144 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
113, 10bitr4d 189 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  ∃!wreu 2355  crio 5518  (class class class)co 5563  cc 7076   + caddc 7081  cmin 7381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308  ax-resscn 7165  ax-1cn 7166  ax-icn 7168  ax-addcl 7169  ax-addrcl 7170  ax-mulcl 7171  ax-addcom 7173  ax-addass 7175  ax-distr 7177  ax-i2m1 7178  ax-0id 7181  ax-rnegex 7182  ax-cnre 7184
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2611  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-sub 7383
This theorem is referenced by:  subadd2  7414  subsub23  7415  pncan  7416  pncan3  7418  addsubeq4  7425  subsub2  7438  renegcl  7471  subaddi  7497  subaddd  7539  fzen  9173  nn0ennn  9550  odd2np1  10463  divalgb  10515
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