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Theorem riotaprop 5415
Description: Properties of a restricted definite description operator. Todo (df-riota 5393 update): can some uses of riota2f 5413 be shortened with this? (Contributed by NM, 23-Nov-2013.)
Hypotheses
Ref Expression
riotaprop.0 xψ
riotaprop.1 B = (x A φ)
riotaprop.2 (x = B → (φψ))
Assertion
Ref Expression
riotaprop (∃!x A φ → (B A ψ))
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 = (x A φ)
2 riotacl 5406 . . 3 (∃!x A φ → (x A φ) A)
31, 2syl5eqel 2106 . 2 (∃!x A φB A)
41eqcomi 2026 . . . 4 (x A φ) = B
5 nfriota1 5399 . . . . . 6 x(x A φ)
61, 5nfcxfr 2157 . . . . 5 xB
7 riotaprop.0 . . . . 5 xψ
8 riotaprop.2 . . . . 5 (x = B → (φψ))
96, 7, 8riota2f 5413 . . . 4 ((B A ∃!x A φ) → (ψ ↔ (x A φ) = B))
104, 9mpbiri 157 . . 3 ((B A ∃!x A φ) → ψ)
113, 10mpancom 401 . 2 (∃!x A φψ)
123, 11jca 290 1 (∃!x A φ → (B A ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wnf 1329   wcel 1374  ∃!wreu 2286  crio 5392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794  df-riota 5393
This theorem is referenced by: (None)
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