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Theorem riota5f 5871
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 2563 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝐵))
3 riota5f.2 . . . 4 (𝜑𝐵𝐴)
4 trud 1380 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → ⊤)
5 reu6i 2943 . . . . . . . . 9 ((𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦)) → ∃!𝑥𝐴 𝜓)
65adantl 277 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → ∃!𝑥𝐴 𝜓)
7 nfv 1539 . . . . . . . . . 10 𝑥𝜑
8 nfv 1539 . . . . . . . . . . 11 𝑥 𝑦𝐴
9 nfra1 2521 . . . . . . . . . . 11 𝑥𝑥𝐴 (𝜓𝑥 = 𝑦)
108, 9nfan 1576 . . . . . . . . . 10 𝑥(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
117, 10nfan 1576 . . . . . . . . 9 𝑥(𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
12 nfcvd 2333 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → 𝑥𝑦)
13 nfvd 1540 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → Ⅎ𝑥⊤)
14 simprl 529 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → 𝑦𝐴)
15 simpr 110 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
16 simplrr 536 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
17 simplrl 535 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑦𝐴)
1815, 17eqeltrd 2266 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥𝐴)
19 rsp 2537 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 → (𝜓𝑥 = 𝑦)))
2016, 18, 19sylc 62 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓𝑥 = 𝑦))
2115, 20mpbird 167 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝜓)
22 trud 1380 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ⊤)
2321, 222thd 175 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ ⊤))
2411, 12, 13, 14, 23riota2df 5867 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ ∃!𝑥𝐴 𝜓) → (⊤ ↔ (𝑥𝐴 𝜓) = 𝑦))
256, 24mpdan 421 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → (⊤ ↔ (𝑥𝐴 𝜓) = 𝑦))
264, 25mpbid 147 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → (𝑥𝐴 𝜓) = 𝑦)
2726expr 375 . . . . 5 ((𝜑𝑦𝐴) → (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
2827ralrimiva 2563 . . . 4 (𝜑 → ∀𝑦𝐴 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
29 rspsbc 3060 . . . 4 (𝐵𝐴 → (∀𝑦𝐴 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) → [𝐵 / 𝑦](∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦)))
303, 28, 29sylc 62 . . 3 (𝜑[𝐵 / 𝑦](∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
31 nfcvd 2333 . . . . . . . 8 (𝜑𝑥𝑦)
32 riota5f.1 . . . . . . . 8 (𝜑𝑥𝐵)
3331, 32nfeqd 2347 . . . . . . 7 (𝜑 → Ⅎ𝑥 𝑦 = 𝐵)
347, 33nfan1 1575 . . . . . 6 𝑥(𝜑𝑦 = 𝐵)
35 simpr 110 . . . . . . . 8 ((𝜑𝑦 = 𝐵) → 𝑦 = 𝐵)
3635eqeq2d 2201 . . . . . . 7 ((𝜑𝑦 = 𝐵) → (𝑥 = 𝑦𝑥 = 𝐵))
3736bibi2d 232 . . . . . 6 ((𝜑𝑦 = 𝐵) → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐵)))
3834, 37ralbid 2488 . . . . 5 ((𝜑𝑦 = 𝐵) → (∀𝑥𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜓𝑥 = 𝐵)))
3935eqeq2d 2201 . . . . 5 ((𝜑𝑦 = 𝐵) → ((𝑥𝐴 𝜓) = 𝑦 ↔ (𝑥𝐴 𝜓) = 𝐵))
4038, 39imbi12d 234 . . . 4 ((𝜑𝑦 = 𝐵) → ((∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) ↔ (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵)))
413, 40sbcied 3014 . . 3 (𝜑 → ([𝐵 / 𝑦](∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) ↔ (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵)))
4230, 41mpbid 147 . 2 (𝜑 → (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵))
432, 42mpd 13 1 (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wtru 1365  wcel 2160  wnfc 2319  wral 2468  ∃!wreu 2470  [wsbc 2977  crio 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-v 2754  df-sbc 2978  df-un 3148  df-sn 3613  df-pr 3614  df-uni 3825  df-iota 5193  df-riota 5847
This theorem is referenced by:  riota5  5872
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