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Theorem iotavalb 43867
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6519. (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 6519 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2 iotasbc 43856 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3 iotaexeu 43855 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4 eqsbc1 3826 . . . . 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 2022 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 342 . . . . . 6 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1916 . . . . 5 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
109biimpac 478 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
1110exlimiv 1926 . . 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 1532   = wceq 1534  wex 1774  wcel 2099  ∃!weu 2558  Vcvv 3471  [wsbc 3776  cio 6498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-sbc 3777  df-un 3952  df-in 3954  df-ss 3964  df-sn 4630  df-pr 4632  df-uni 4909  df-iota 6500
This theorem is referenced by:  iotavalsb  43870
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