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Theorem resubval 39453
 Description: Value of real subtraction, which is the (unique) real 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by Steven Nguyen, 7-Jan-2022.)
Assertion
Ref Expression
resubval ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem resubval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2834 . . 3 (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
21riotabidv 7100 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦) = (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴))
3 oveq1 7147 . . . 4 (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
43eqeq1d 2824 . . 3 (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
54riotabidv 7100 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
6 df-resub 39452 . 2 = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦))
7 riotaex 7102 . 2 (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) ∈ V
82, 5, 6, 7ovmpo 7294 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ℩crio 7097  (class class class)co 7140  ℝcr 10525   + caddc 10529   −ℝ cresub 39451 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-resub 39452 This theorem is referenced by:  rernegcl  39457  renegadd  39458  rersubcl  39464  resubadd  39465  resubf  39467  resubeqsub  39513
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