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Theorem iotaval 5067
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 5057 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
2 vex 2661 . . . . . . 7 𝑦 ∈ V
3 sbeqalb 2935 . . . . . . . 8 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧))
4 equcomi 1663 . . . . . . . 8 (𝑦 = 𝑧𝑧 = 𝑦)
53, 4syl6 33 . . . . . . 7 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑧 = 𝑦))
62, 5ax-mp 5 . . . . . 6 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑧 = 𝑦)
76ex 114 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = 𝑦))
8 equequ2 1672 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
98equcoms 1667 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
109bibi2d 231 . . . . . . . 8 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
1110biimpd 143 . . . . . . 7 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑧)))
1211alimdv 1833 . . . . . 6 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑧)))
1312com12 30 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑧 = 𝑦 → ∀𝑥(𝜑𝑥 = 𝑧)))
147, 13impbid 128 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
1514alrimiv 1828 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
16 uniabio 5066 . . 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 2160 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312   = wceq 1314  wcel 1463  {cab 2101  Vcvv 2658   cuni 3704  cio 5054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-sn 3501  df-pr 3502  df-uni 3705  df-iota 5056
This theorem is referenced by:  iotauni  5068  iota1  5070  euiotaex  5072  iota4  5074  iota5  5076
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