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Theorem riota5f 7396
Description: A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota5f.1 (𝜑𝑥𝐵)
riota5f.2 (𝜑𝐵𝐴)
riota5f.3 ((𝜑𝑥𝐴) → (𝜓𝑥 = 𝐵))
Assertion
Ref Expression
riota5f (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem riota5f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 riota5f.3 . . 3 ((𝜑𝑥𝐴) → (𝜓𝑥 = 𝐵))
21ralrimiva 3146 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝐵))
3 riota5f.2 . . . 4 (𝜑𝐵𝐴)
4 trud 1551 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → ⊤)
5 reu6i 3724 . . . . . . . . 9 ((𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦)) → ∃!𝑥𝐴 𝜓)
65adantl 482 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → ∃!𝑥𝐴 𝜓)
7 nfv 1917 . . . . . . . . . 10 𝑥𝜑
8 nfv 1917 . . . . . . . . . . 11 𝑥 𝑦𝐴
9 nfra1 3281 . . . . . . . . . . 11 𝑥𝑥𝐴 (𝜓𝑥 = 𝑦)
108, 9nfan 1902 . . . . . . . . . 10 𝑥(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
117, 10nfan 1902 . . . . . . . . 9 𝑥(𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
12 nfcvd 2904 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → 𝑥𝑦)
13 nfvd 1918 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → Ⅎ𝑥⊤)
14 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → 𝑦𝐴)
15 simpr 485 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
16 simplrr 776 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
17 simplrl 775 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑦𝐴)
1815, 17eqeltrd 2833 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥𝐴)
19 rsp 3244 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 → (𝜓𝑥 = 𝑦)))
2016, 18, 19sylc 65 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓𝑥 = 𝑦))
2115, 20mpbird 256 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝜓)
22 trud 1551 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ⊤)
2321, 222thd 264 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ ⊤))
2411, 12, 13, 14, 23riota2df 7391 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ ∃!𝑥𝐴 𝜓) → (⊤ ↔ (𝑥𝐴 𝜓) = 𝑦))
256, 24mpdan 685 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → (⊤ ↔ (𝑥𝐴 𝜓) = 𝑦))
264, 25mpbid 231 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → (𝑥𝐴 𝜓) = 𝑦)
2726expr 457 . . . . 5 ((𝜑𝑦𝐴) → (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
2827ralrimiva 3146 . . . 4 (𝜑 → ∀𝑦𝐴 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
29 rspsbc 3873 . . . 4 (𝐵𝐴 → (∀𝑦𝐴 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) → [𝐵 / 𝑦](∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦)))
303, 28, 29sylc 65 . . 3 (𝜑[𝐵 / 𝑦](∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
31 nfcvd 2904 . . . . . . . 8 (𝜑𝑥𝑦)
32 riota5f.1 . . . . . . . 8 (𝜑𝑥𝐵)
3331, 32nfeqd 2913 . . . . . . 7 (𝜑 → Ⅎ𝑥 𝑦 = 𝐵)
347, 33nfan1 2193 . . . . . 6 𝑥(𝜑𝑦 = 𝐵)
35 simpr 485 . . . . . . . 8 ((𝜑𝑦 = 𝐵) → 𝑦 = 𝐵)
3635eqeq2d 2743 . . . . . . 7 ((𝜑𝑦 = 𝐵) → (𝑥 = 𝑦𝑥 = 𝐵))
3736bibi2d 342 . . . . . 6 ((𝜑𝑦 = 𝐵) → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐵)))
3834, 37ralbid 3270 . . . . 5 ((𝜑𝑦 = 𝐵) → (∀𝑥𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜓𝑥 = 𝐵)))
3935eqeq2d 2743 . . . . 5 ((𝜑𝑦 = 𝐵) → ((𝑥𝐴 𝜓) = 𝑦 ↔ (𝑥𝐴 𝜓) = 𝐵))
4038, 39imbi12d 344 . . . 4 ((𝜑𝑦 = 𝐵) → ((∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) ↔ (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵)))
413, 40sbcied 3822 . . 3 (𝜑 → ([𝐵 / 𝑦](∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) ↔ (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵)))
4230, 41mpbid 231 . 2 (𝜑 → (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵))
432, 42mpd 15 1 (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wtru 1542  wcel 2106  wnfc 2883  wral 3061  ∃!wreu 3374  [wsbc 3777  crio 7366
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 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-reu 3377  df-v 3476  df-sbc 3778  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-riota 7367
This theorem is referenced by:  riota5  7397
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