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Theorem iotaval 5765
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 5755 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
2 vex 3175 . . . . . . 7 𝑦 ∈ V
3 sbeqalb 3454 . . . . . . . 8 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧))
4 equcomi 1930 . . . . . . . 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 448 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = 𝑦))
8 equequ2 1939 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
98equcoms 1933 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
109bibi2d 330 . . . . . . . 8 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
1110biimpd 217 . . . . . . 7 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑧)))
1211alimdv 1831 . . . . . 6 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑧)))
1312com12 32 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑧 = 𝑦 → ∀𝑥(𝜑𝑥 = 𝑧)))
147, 13impbid 200 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
1514alrimiv 1841 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
16 uniabio 5764 . . 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 2655 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wcel 1976  {cab 2595  Vcvv 3172   cuni 4366  cio 5752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-v 3174  df-sbc 3402  df-un 3544  df-sn 4125  df-pr 4127  df-uni 4367  df-iota 5754
This theorem is referenced by:  iotauni  5766  iota1  5768  iotaex  5771  iota4  5772  iota5  5774  iota5f  30667  iotain  37443  iotaexeu  37444  iotasbc  37445  iotaequ  37455  iotavalb  37456  pm14.24  37458  sbiota1  37460
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