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Theorem iotavalOLD 6537
Description: Obsolete version of iotaval 6534 as of 23-Dec-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
iotavalOLD (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotavalOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiota2 6517 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
2 sbeqalb 3859 . . . . . . . 8 (𝑦 ∈ V → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧))
32elv 3483 . . . . . . 7 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜑𝑥 = 𝑧)) → 𝑦 = 𝑧)
43ex 412 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) → 𝑦 = 𝑧))
5 equequ2 2023 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65bibi2d 342 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
76biimpd 229 . . . . . . . 8 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑧)))
87alimdv 1914 . . . . . . 7 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑧)))
98com12 32 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑦 = 𝑧 → ∀𝑥(𝜑𝑥 = 𝑧)))
104, 9impbid 212 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑦 = 𝑧))
11 equcom 2015 . . . . 5 (𝑦 = 𝑧𝑧 = 𝑦)
1210, 11bitrdi 287 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
1312alrimiv 1925 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
14 uniabio 6530 . . 3 (∀𝑧(∀𝑥(𝜑𝑥 = 𝑧) ↔ 𝑧 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
1513, 14syl 17 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = 𝑦)
161, 15eqtrid 2787 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  {cab 2712  Vcvv 3478   cuni 4912  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by: (None)
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