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Theorem riota5f 7116
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 3170 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝑥 = 𝐵))
3 riota5f.2 . . . 4 (𝜑𝐵𝐴)
4 trud 1548 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → ⊤)
5 reu6i 3696 . . . . . . . . 9 ((𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦)) → ∃!𝑥𝐴 𝜓)
65adantl 485 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → ∃!𝑥𝐴 𝜓)
7 nfv 1916 . . . . . . . . . 10 𝑥𝜑
8 nfv 1916 . . . . . . . . . . 11 𝑥 𝑦𝐴
9 nfra1 3207 . . . . . . . . . . 11 𝑥𝑥𝐴 (𝜓𝑥 = 𝑦)
108, 9nfan 1901 . . . . . . . . . 10 𝑥(𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
117, 10nfan 1901 . . . . . . . . 9 𝑥(𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦)))
12 nfcvd 2975 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → 𝑥𝑦)
13 nfvd 1917 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → Ⅎ𝑥⊤)
14 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → 𝑦𝐴)
15 simpr 488 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
16 simplrr 777 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ∀𝑥𝐴 (𝜓𝑥 = 𝑦))
17 simplrl 776 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑦𝐴)
1815, 17eqeltrd 2912 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥𝐴)
19 rsp 3193 . . . . . . . . . . . 12 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 → (𝜓𝑥 = 𝑦)))
2016, 18, 19sylc 65 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓𝑥 = 𝑦))
2115, 20mpbird 260 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝜓)
22 trud 1548 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ⊤)
2321, 222thd 268 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ ⊤))
2411, 12, 13, 14, 23riota2df 7111 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) ∧ ∃!𝑥𝐴 𝜓) → (⊤ ↔ (𝑥𝐴 𝜓) = 𝑦))
256, 24mpdan 686 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → (⊤ ↔ (𝑥𝐴 𝜓) = 𝑦))
264, 25mpbid 235 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑦))) → (𝑥𝐴 𝜓) = 𝑦)
2726expr 460 . . . . 5 ((𝜑𝑦𝐴) → (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
2827ralrimiva 3170 . . . 4 (𝜑 → ∀𝑦𝐴 (∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦))
29 rspsbc 3837 . . . 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 2975 . . . . . . . 8 (𝜑𝑥𝑦)
32 riota5f.1 . . . . . . . 8 (𝜑𝑥𝐵)
3331, 32nfeqd 2984 . . . . . . 7 (𝜑 → Ⅎ𝑥 𝑦 = 𝐵)
347, 33nfan1 2201 . . . . . 6 𝑥(𝜑𝑦 = 𝐵)
35 simpr 488 . . . . . . . 8 ((𝜑𝑦 = 𝐵) → 𝑦 = 𝐵)
3635eqeq2d 2832 . . . . . . 7 ((𝜑𝑦 = 𝐵) → (𝑥 = 𝑦𝑥 = 𝐵))
3736bibi2d 346 . . . . . 6 ((𝜑𝑦 = 𝐵) → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐵)))
3834, 37ralbid 3219 . . . . 5 ((𝜑𝑦 = 𝐵) → (∀𝑥𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜓𝑥 = 𝐵)))
3935eqeq2d 2832 . . . . 5 ((𝜑𝑦 = 𝐵) → ((𝑥𝐴 𝜓) = 𝑦 ↔ (𝑥𝐴 𝜓) = 𝐵))
4038, 39imbi12d 348 . . . 4 ((𝜑𝑦 = 𝐵) → ((∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) ↔ (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵)))
413, 40sbcied 3791 . . 3 (𝜑 → ([𝐵 / 𝑦](∀𝑥𝐴 (𝜓𝑥 = 𝑦) → (𝑥𝐴 𝜓) = 𝑦) ↔ (∀𝑥𝐴 (𝜓𝑥 = 𝐵) → (𝑥𝐴 𝜓) = 𝐵)))
4230, 41mpbid 235 . 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 209  wa 399   = wceq 1538  wtru 1539  wcel 2115  wnfc 2958  wral 3126  ∃!wreu 3128  [wsbc 3749  crio 7087
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-reu 3133  df-v 3473  df-sbc 3750  df-un 3915  df-in 3917  df-ss 3927  df-sn 4541  df-pr 4543  df-uni 4812  df-iota 6287  df-riota 7088
This theorem is referenced by:  riota5  7117
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