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Theorem iotavalb 38948
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5900. (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 5900 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2 iotasbc 38937 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3 iotaexeu 38936 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4 eqsbc3 3508 . . . . 5 ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
53, 4syl 17 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
62, 5bitr3d 270 . . 3 (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
7 equequ2 1999 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87bibi2d 331 . . . . . 6 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1889 . . . . 5 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
109biimpac 502 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
1110exlimiv 1898 . . 3 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
126, 11syl6bir 244 . 2 (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑𝑥 = 𝑦)))
131, 12impbid2 216 1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  Vcvv 3231  [wsbc 3468  cio 5887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-sbc 3469  df-un 3612  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889
This theorem is referenced by:  iotavalsb  38951
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