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Theorem riotaprop 6799
Description: Properties of a restricted definite description operator. TODO (df-riota 6775 update): can some uses of riota2f 6796 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 6789 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
31, 2syl5eqel 2843 . 2 (∃!𝑥𝐴 𝜑𝐵𝐴)
41eqcomi 2769 . . . 4 (𝑥𝐴 𝜑) = 𝐵
5 nfriota1 6782 . . . . . 6 𝑥(𝑥𝐴 𝜑)
61, 5nfcxfr 2900 . . . . 5 𝑥𝐵
7 riotaprop.0 . . . . 5 𝑥𝜓
8 riotaprop.2 . . . . 5 (𝑥 = 𝐵 → (𝜑𝜓))
96, 7, 8riota2f 6796 . . . 4 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
104, 9mpbiri 248 . . 3 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → 𝜓)
113, 10mpancom 706 . 2 (∃!𝑥𝐴 𝜑𝜓)
123, 11jca 555 1 (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wnf 1857  wcel 2139  ∃!wreu 3052  crio 6774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-un 3720  df-in 3722  df-ss 3729  df-sn 4322  df-pr 4324  df-uni 4589  df-iota 6012  df-riota 6775
This theorem is referenced by:  fin23lem27  9362  lble  11187  ltrniotaval  36389
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