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Theorem iotavalb 39307
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6044. (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 6044 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2 iotasbc 39296 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3 iotaexeu 39295 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4 eqsbc3 3638 . . . . 5 ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
53, 4syl 17 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
62, 5bitr3d 272 . . 3 (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
7 equequ2 2123 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 333 . . . . . 6 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 2015 . . . . 5 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
109biimpac 470 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
1110exlimiv 2025 . . 3 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
126, 11syl6bir 245 . 2 (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑𝑥 = 𝑦)))
131, 12impbid2 217 1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  ∃!weu 2581  Vcvv 3350  [wsbc 3598  cio 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-v 3352  df-sbc 3599  df-un 3739  df-sn 4337  df-pr 4339  df-uni 4597  df-iota 6033
This theorem is referenced by:  iotavalsb  39310
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