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Theorem subadd 8159
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 8148 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
21eqeq1d 2186 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
323adant3 1017 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
4 negeu 8147 . . . . 5 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
5 oveq2 5882 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 + 𝑥) = (𝐵 + 𝐶))
65eqeq1d 2186 . . . . . 6 (𝑥 = 𝐶 → ((𝐵 + 𝑥) = 𝐴 ↔ (𝐵 + 𝐶) = 𝐴))
76riota2 5852 . . . . 5 ((𝐶 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
84, 7sylan2 286 . . . 4 ((𝐶 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
983impb 1199 . . 3 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
1093com13 1208 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 + 𝐶) = 𝐴 ↔ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) = 𝐶))
113, 10bitr4d 191 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  ∃!wreu 2457  crio 5829  (class class class)co 5874  cc 7808   + caddc 7813  cmin 8127
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-setind 4536  ax-resscn 7902  ax-1cn 7903  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-sub 8129
This theorem is referenced by:  subadd2  8160  subsub23  8161  pncan  8162  pncan3  8164  addsubeq4  8171  subsub2  8184  renegcl  8217  subaddi  8243  subaddd  8285  fzen  10042  nn0ennn  10432  cos2t  11757  cos2tsin  11758  odd2np1  11877  divalgb  11929  sincosq1eq  14230  coskpi  14239
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