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Theorem resubval 42397
Description: Value of real subtraction, which is the (unique) real 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by Steven Nguyen, 7-Jan-2023.)
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 2749 . . 3 (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
21riotabidv 7390 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦) = (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴))
3 oveq1 7438 . . . 4 (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
43eqeq1d 2739 . . 3 (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
54riotabidv 7390 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
6 df-resub 42396 . 2 = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦))
7 riotaex 7392 . 2 (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) ∈ V
82, 5, 6, 7ovmpo 7593 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  crio 7387  (class class class)co 7431  cr 11154   + caddc 11158   cresub 42395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-resub 42396
This theorem is referenced by:  rernegcl  42401  renegadd  42402  rersubcl  42408  resubadd  42409  resubf  42411  resubeqsub  42459
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