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Theorem iotavalb 43738
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6505. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotavalb (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotavalb
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 iotaval 6505 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2 iotasbc 43727 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3 iotaexeu 43726 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4 eqsbc1 3819 . . . . 5 ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
53, 4syl 17 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
62, 5bitr3d 281 . . 3 (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
7 equequ2 2021 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 342 . . . . . 6 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1915 . . . . 5 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
109biimpac 478 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
1110exlimiv 1925 . . 3 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
126, 11syl6bir 254 . 2 (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑𝑥 = 𝑦)))
131, 12impbid2 225 1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  ∃!weu 2554  Vcvv 3466  [wsbc 3770  cio 6484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-sbc 3771  df-un 3946  df-in 3948  df-ss 3958  df-sn 4622  df-pr 4624  df-uni 4901  df-iota 6486
This theorem is referenced by:  iotavalsb  43741
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