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Theorem iotavalb 41132
 Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6298. (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 6298 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2 iotasbc 41121 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3 iotaexeu 41120 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4 eqsbc3 3765 . . . . 5 ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
53, 4syl 17 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
62, 5bitr3d 284 . . 3 (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
7 equequ2 2033 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 346 . . . . . 6 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1921 . . . . 5 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
109biimpac 482 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
1110exlimiv 1931 . . 3 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
126, 11syl6bir 257 . 2 (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑𝑥 = 𝑦)))
131, 12impbid2 229 1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  Vcvv 3441  [wsbc 3720  ℩cio 6281 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4801  df-iota 6283 This theorem is referenced by:  iotavalsb  41135
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