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Theorem iotaval 6023
 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 6013 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
2 vex 3343 . . . . . . . 8 𝑦 ∈ V
3 sbeqalb 3629 . . . . . . . 8 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧))
42, 3ax-mp 5 . . . . . . 7 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧)
54ex 449 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) → 𝑦 = 𝑧))
6 equequ2 2108 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
76bibi2d 331 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
87biimpd 219 . . . . . . . 8 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑧)))
98alimdv 1994 . . . . . . 7 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑧)))
109com12 32 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑦 = 𝑧 → ∀𝑥(𝜑𝑥 = 𝑧)))
115, 10impbid 202 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑦 = 𝑧))
12 equcom 2100 . . . . 5 (𝑦 = 𝑧𝑧 = 𝑦)
1311, 12syl6bb 276 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
1413alrimiv 2004 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
15 uniabio 6022 . . 3 (∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
1614, 15syl 17 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
171, 16syl5eq 2806 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1630   = wceq 1632   ∈ wcel 2139  {cab 2746  Vcvv 3340  ∪ cuni 4588  ℩cio 6010 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-sbc 3577  df-un 3720  df-sn 4322  df-pr 4324  df-uni 4589  df-iota 6012 This theorem is referenced by:  iotauni  6024  iota1  6026  iotaex  6029  iota4  6030  iota5  6032  iota5f  31913  iotain  39120  iotaexeu  39121  iotasbc  39122  iotaequ  39132  iotavalb  39133  pm14.24  39135  sbiota1  39137
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