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Theorem iotaval 6322
Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaval (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotaval
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiota2 6308 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
2 sbeqalb 3833 . . . . . . . 8 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧))
32elv 3497 . . . . . . 7 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧)
43ex 413 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) → 𝑦 = 𝑧))
5 equequ2 2024 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65bibi2d 344 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
76biimpd 230 . . . . . . . 8 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑧)))
87alimdv 1908 . . . . . . 7 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑧)))
98com12 32 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑦 = 𝑧 → ∀𝑥(𝜑𝑥 = 𝑧)))
104, 9impbid 213 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑦 = 𝑧))
11 equcom 2016 . . . . 5 (𝑦 = 𝑧𝑧 = 𝑦)
1210, 11syl6bb 288 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
1312alrimiv 1919 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
14 uniabio 6321 . . 3 (∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
1513, 14syl 17 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
161, 15syl5eq 2865 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  {cab 2796  Vcvv 3492   cuni 4830  cio 6305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-sbc 3770  df-un 3938  df-sn 4558  df-pr 4560  df-uni 4831  df-iota 6307
This theorem is referenced by:  iotauni  6323  iota1  6325  iotaex  6328  iota4  6329  iota5  6331  iota5f  32852  iotain  40626  iotaexeu  40627  iotasbc  40628  iotaequ  40638  iotavalb  40639  pm14.24  40641  sbiota1  40643  aiotaval  43170
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