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Theorem riotaprop 7400
Description: Properties of a restricted definite description operator. (Contributed by NM, 23-Nov-2013.)
Hypotheses
Ref Expression
riotaprop.0 𝑥𝜓
riotaprop.1 𝐵 = (𝑥𝐴 𝜑)
riotaprop.2 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
riotaprop (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem riotaprop
StepHypRef Expression
1 riotaprop.1 . . 3 𝐵 = (𝑥𝐴 𝜑)
2 riotacl 7390 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
31, 2eqeltrid 2829 . 2 (∃!𝑥𝐴 𝜑𝐵𝐴)
41eqcomi 2734 . . . 4 (𝑥𝐴 𝜑) = 𝐵
5 nfriota1 7379 . . . . . 6 𝑥(𝑥𝐴 𝜑)
61, 5nfcxfr 2890 . . . . 5 𝑥𝐵
7 riotaprop.0 . . . . 5 𝑥𝜓
8 riotaprop.2 . . . . 5 (𝑥 = 𝐵 → (𝜑𝜓))
96, 7, 8riota2f 7397 . . . 4 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
104, 9mpbiri 257 . . 3 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → 𝜓)
113, 10mpancom 686 . 2 (∃!𝑥𝐴 𝜑𝜓)
123, 11jca 510 1 (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wnf 1777  wcel 2098  ∃!wreu 3362  crio 7371
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-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-un 3944  df-ss 3956  df-sn 4625  df-pr 4627  df-uni 4904  df-iota 6495  df-riota 7372
This theorem is referenced by:  fin23lem27  10351  lble  12196  ltrniotaval  40110
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