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Theorem resubval 40822
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 2748 . . 3 (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
21riotabidv 7315 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦) = (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴))
3 oveq1 7364 . . . 4 (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
43eqeq1d 2738 . . 3 (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
54riotabidv 7315 . 2 (𝑧 = 𝐵 → (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
6 df-resub 40821 . 2 = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦))
7 riotaex 7317 . 2 (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) ∈ V
82, 5, 6, 7ovmpo 7515 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  crio 7312  (class class class)co 7357  cr 11050   + caddc 11054   cresub 40820
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-resub 40821
This theorem is referenced by:  rernegcl  40826  renegadd  40827  rersubcl  40833  resubadd  40834  resubf  40836  resubeqsub  40884
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