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Theorem iota5f 32121
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 2956 . . . 4 𝑥 𝐴𝑉
41, 3nfan 1999 . . 3 𝑥(𝜑𝐴𝑉)
5 iota5f.3 . . 3 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
64, 5alrimi 2248 . 2 ((𝜑𝐴𝑉) → ∀𝑥(𝜓𝑥 = 𝐴))
72nfeq2 2957 . . . . . 6 𝑥 𝑦 = 𝐴
8 eqeq2 2810 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
98bibi2d 334 . . . . . 6 (𝑦 = 𝐴 → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐴)))
107, 9albid 2257 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜓𝑥 = 𝐴)))
11 eqeq2 2810 . . . . 5 (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
1210, 11imbi12d 336 . . . 4 (𝑦 = 𝐴 → ((∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
13 iotaval 6075 . . . 4 (∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
1412, 13vtoclg 3453 . . 3 (𝐴𝑉 → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
1514adantl 474 . 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 198  wa 385  wal 1651   = wceq 1653  wnf 1879  wcel 2157  wnfc 2928  cio 6062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-v 3387  df-sbc 3634  df-un 3774  df-sn 4369  df-pr 4371  df-uni 4629  df-iota 6064
This theorem is referenced by: (None)
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