ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  riota5f GIF version

Theorem riota5f 5831
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 2543 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝐵))
3 riota5f.2 . . . 4 (𝜑𝐵𝐴)
4 a1tru 1364 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → ⊤)
5 reu6i 2921 . . . . . . . . 9 ((𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦)) → ∃!𝑥𝐴 𝜓)
65adantl 275 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → ∃!𝑥𝐴 𝜓)
7 nfv 1521 . . . . . . . . . 10 𝑥𝜑
8 nfv 1521 . . . . . . . . . . 11 𝑥 𝑦𝐴
9 nfra1 2501 . . . . . . . . . . 11 𝑥𝑥𝐴 (𝜓𝑥 = 𝑦)
108, 9nfan 1558 . . . . . . . . . 10 𝑥(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
117, 10nfan 1558 . . . . . . . . 9 𝑥(𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
12 nfcvd 2313 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → 𝑥𝑦)
13 nfvd 1522 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → Ⅎ𝑥⊤)
14 simprl 526 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → 𝑦𝐴)
15 simpr 109 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
16 simplrr 531 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
17 simplrl 530 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑦𝐴)
1815, 17eqeltrd 2247 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥𝐴)
19 rsp 2517 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 → (𝜓𝑥 = 𝑦)))
2016, 18, 19sylc 62 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓𝑥 = 𝑦))
2115, 20mpbird 166 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝜓)
22 a1tru 1364 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ⊤)
2321, 222thd 174 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ ⊤))
2411, 12, 13, 14, 23riota2df 5827 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ ∃!𝑥𝐴 𝜓) → (⊤ ↔ (𝑥𝐴 𝜓) = 𝑦))
256, 24mpdan 419 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → (⊤ ↔ (𝑥𝐴 𝜓) = 𝑦))
264, 25mpbid 146 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → (𝑥𝐴 𝜓) = 𝑦)
2726expr 373 . . . . 5 ((𝜑𝑦𝐴) → (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
2827ralrimiva 2543 . . . 4 (𝜑 → ∀𝑦𝐴 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
29 rspsbc 3037 . . . 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 2313 . . . . . . . 8 (𝜑𝑥𝑦)
32 riota5f.1 . . . . . . . 8 (𝜑𝑥𝐵)
3331, 32nfeqd 2327 . . . . . . 7 (𝜑 → Ⅎ𝑥 𝑦 = 𝐵)
347, 33nfan1 1557 . . . . . 6 𝑥(𝜑𝑦 = 𝐵)
35 simpr 109 . . . . . . . 8 ((𝜑𝑦 = 𝐵) → 𝑦 = 𝐵)
3635eqeq2d 2182 . . . . . . 7 ((𝜑𝑦 = 𝐵) → (𝑥 = 𝑦𝑥 = 𝐵))
3736bibi2d 231 . . . . . 6 ((𝜑𝑦 = 𝐵) → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐵)))
3834, 37ralbid 2468 . . . . 5 ((𝜑𝑦 = 𝐵) → (∀𝑥𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜓𝑥 = 𝐵)))
3935eqeq2d 2182 . . . . 5 ((𝜑𝑦 = 𝐵) → ((𝑥𝐴 𝜓) = 𝑦 ↔ (𝑥𝐴 𝜓) = 𝐵))
4038, 39imbi12d 233 . . . 4 ((𝜑𝑦 = 𝐵) → ((∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) ↔ (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵)))
413, 40sbcied 2991 . . 3 (𝜑 → ([𝐵 / 𝑦](∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) ↔ (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵)))
4230, 41mpbid 146 . 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 103  wb 104   = wceq 1348  wtru 1349  wcel 2141  wnfc 2299  wral 2448  ∃!wreu 2450  [wsbc 2955  crio 5806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3587  df-pr 3588  df-uni 3795  df-iota 5158  df-riota 5807
This theorem is referenced by:  riota5  5832
  Copyright terms: Public domain W3C validator