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Theorem iotaval 5779
 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 5769 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
2 vex 3176 . . . . . . 7 𝑦 ∈ V
3 sbeqalb 3455 . . . . . . . 8 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧))
4 equcomi 1931 . . . . . . . 8 (𝑦 = 𝑧𝑧 = 𝑦)
53, 4syl6 34 . . . . . . 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 449 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = 𝑦))
8 equequ2 1940 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
98equcoms 1934 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
109bibi2d 331 . . . . . . . 8 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
1110biimpd 218 . . . . . . 7 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑧)))
1211alimdv 1832 . . . . . 6 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑧)))
1312com12 32 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑧 = 𝑦 → ∀𝑥(𝜑𝑥 = 𝑧)))
147, 13impbid 201 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
1514alrimiv 1842 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
16 uniabio 5778 . . 3 (∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
1715, 16syl 17 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
181, 17syl5eq 2656 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  ∪ cuni 4372  ℩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:  iotauni  5780  iota1  5782  iotaex  5785  iota4  5786  iota5  5788  iota5f  30861  iotain  37640  iotaexeu  37641  iotasbc  37642  iotaequ  37652  iotavalb  37653  pm14.24  37655  sbiota1  37657
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