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Theorem riotaprop 7392
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 7382 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
31, 2eqeltrid 2837 . 2 (∃!𝑥𝐴 𝜑𝐵𝐴)
41eqcomi 2741 . . . 4 (𝑥𝐴 𝜑) = 𝐵
5 nfriota1 7371 . . . . . 6 𝑥(𝑥𝐴 𝜑)
61, 5nfcxfr 2901 . . . . 5 𝑥𝐵
7 riotaprop.0 . . . . 5 𝑥𝜓
8 riotaprop.2 . . . . 5 (𝑥 = 𝐵 → (𝜑𝜓))
96, 7, 8riota2f 7389 . . . 4 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
104, 9mpbiri 257 . . 3 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → 𝜓)
113, 10mpancom 686 . 2 (∃!𝑥𝐴 𝜑𝜓)
123, 11jca 512 1 (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wnf 1785  wcel 2106  ∃!wreu 3374  crio 7363
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-riota 7364
This theorem is referenced by:  fin23lem27  10322  lble  12165  ltrniotaval  39447
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