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Theorem riotaprop 6600
Description: Properties of a restricted definite description operator. TODO (df-riota 6576 update): can some uses of riota2f 6597 be shortened with this? (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 6590 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
31, 2syl5eqel 2702 . 2 (∃!𝑥𝐴 𝜑𝐵𝐴)
41eqcomi 2630 . . . 4 (𝑥𝐴 𝜑) = 𝐵
5 nfriota1 6583 . . . . . 6 𝑥(𝑥𝐴 𝜑)
61, 5nfcxfr 2759 . . . . 5 𝑥𝐵
7 riotaprop.0 . . . . 5 𝑥𝜓
8 riotaprop.2 . . . . 5 (𝑥 = 𝐵 → (𝜑𝜓))
96, 7, 8riota2f 6597 . . . 4 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
104, 9mpbiri 248 . . 3 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → 𝜓)
113, 10mpancom 702 . 2 (∃!𝑥𝐴 𝜑𝜓)
123, 11jca 554 1 (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wnf 1705  wcel 1987  ∃!wreu 2910  crio 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-un 3565  df-in 3567  df-ss 3574  df-sn 4156  df-pr 4158  df-uni 4410  df-iota 5820  df-riota 6576
This theorem is referenced by:  fin23lem27  9110  lble  10935  ltrniotaval  35388
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