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Theorem subval 10869
 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 2831 . . 3 (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
21riotabidv 7108 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
3 oveq1 7155 . . . 4 (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
43eqeq1d 2821 . . 3 (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
54riotabidv 7108 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
6 df-sub 10864 . 2 − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
7 riotaex 7110 . 2 (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ V
82, 5, 6, 7ovmpo 7302 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ℩crio 7105  (class class class)co 7148  ℂcc 10527   + caddc 10532   − cmin 10862 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-sub 10864 This theorem is referenced by:  subcl  10877  subf  10880  subadd  10881
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