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Theorem iota5f 30861
 Description: A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
Hypotheses
Ref Expression
iota5f.1 𝑥𝜑
iota5f.2 𝑥𝐴
iota5f.3 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
Assertion
Ref Expression
iota5f ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
Distinct variable group:   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem iota5f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iota5f.1 . . . 4 𝑥𝜑
2 iota5f.2 . . . . 5 𝑥𝐴
32nfel1 2765 . . . 4 𝑥 𝐴𝑉
41, 3nfan 1816 . . 3 𝑥(𝜑𝐴𝑉)
5 iota5f.3 . . 3 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
64, 5alrimi 2069 . 2 ((𝜑𝐴𝑉) → ∀𝑥(𝜓𝑥 = 𝐴))
72nfeq2 2766 . . . . . 6 𝑥 𝑦 = 𝐴
8 eqeq2 2621 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
98bibi2d 331 . . . . . 6 (𝑦 = 𝐴 → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐴)))
107, 9albid 2077 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜓𝑥 = 𝐴)))
11 eqeq2 2621 . . . . 5 (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
1210, 11imbi12d 333 . . . 4 (𝑦 = 𝐴 → ((∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
13 iotaval 5779 . . . 4 (∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
1412, 13vtoclg 3239 . . 3 (𝐴𝑉 → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
1514adantl 481 . 2 ((𝜑𝐴𝑉) → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
166, 15mpd 15 1 ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768 This theorem is referenced by: (None)
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