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Theorem iotaval 4928
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 4918 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
2 vex 2613 . . . . . . 7 𝑦 ∈ V
3 sbeqalb 2879 . . . . . . . 8 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧))
4 equcomi 1633 . . . . . . . 8 (𝑦 = 𝑧𝑧 = 𝑦)
53, 4syl6 33 . . . . . . 7 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑧 = 𝑦))
62, 5ax-mp 7 . . . . . 6 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑧 = 𝑦)
76ex 113 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = 𝑦))
8 equequ2 1641 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
98equcoms 1636 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
109bibi2d 230 . . . . . . . 8 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
1110biimpd 142 . . . . . . 7 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑧)))
1211alimdv 1802 . . . . . 6 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑧)))
1312com12 30 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑧 = 𝑦 → ∀𝑥(𝜑𝑥 = 𝑧)))
147, 13impbid 127 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
1514alrimiv 1797 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
16 uniabio 4927 . . 3 (∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
1715, 16syl 14 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
181, 17syl5eq 2127 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wcel 1434  {cab 2069  Vcvv 2610   cuni 3621  cio 4915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-sn 3422  df-pr 3423  df-uni 3622  df-iota 4917
This theorem is referenced by:  iotauni  4929  iota1  4931  euiotaex  4933  iota4  4935  iota5  4937
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