MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotaprop Structured version   Visualization version   GIF version

Theorem riotaprop 7342
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 7332 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
31, 2eqeltrid 2842 . 2 (∃!𝑥𝐴 𝜑𝐵𝐴)
41eqcomi 2746 . . . 4 (𝑥𝐴 𝜑) = 𝐵
5 nfriota1 7321 . . . . . 6 𝑥(𝑥𝐴 𝜑)
61, 5nfcxfr 2906 . . . . 5 𝑥𝐵
7 riotaprop.0 . . . . 5 𝑥𝜓
8 riotaprop.2 . . . . 5 (𝑥 = 𝐵 → (𝜑𝜓))
96, 7, 8riota2f 7339 . . . 4 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
104, 9mpbiri 258 . . 3 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → 𝜓)
113, 10mpancom 687 . 2 (∃!𝑥𝐴 𝜑𝜓)
123, 11jca 513 1 (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  ∃!wreu 3352  crio 7313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-un 3916  df-in 3918  df-ss 3928  df-sn 4588  df-pr 4590  df-uni 4867  df-iota 6449  df-riota 7314
This theorem is referenced by:  fin23lem27  10265  lble  12108  ltrniotaval  39047
  Copyright terms: Public domain W3C validator