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Theorem subval 10119
Description: Value of subtraction, which is the (unique) element 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
Assertion
Ref Expression
subval ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem subval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2616 . . 3 (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
21riotabidv 6487 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
3 oveq1 6530 . . . 4 (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
43eqeq1d 2607 . . 3 (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
54riotabidv 6487 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
6 df-sub 10115 . 2 − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
7 riotaex 6489 . 2 (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ V
82, 5, 6, 7ovmpt2 6668 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  crio 6484  (class class class)co 6523  cc 9786   + caddc 9791  cmin 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-sub 10115
This theorem is referenced by:  subcl  10127  subf  10130  subadd  10131
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