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Theorem riotaprop 5434
 Description: Properties of a restricted definite description operator. Todo (df-riota 5411 update): can some uses of riota2f 5432 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 5425 . . 3 (∃!x A φ → (x A φ) A)
31, 2syl5eqel 2121 . 2 (∃!x A φB A)
41eqcomi 2041 . . . 4 (x A φ) = B
5 nfriota1 5418 . . . . . 6 x(x A φ)
61, 5nfcxfr 2172 . . . . 5 xB
7 riotaprop.0 . . . . 5 xψ
8 riotaprop.2 . . . . 5 (x = B → (φψ))
96, 7, 8riota2f 5432 . . . 4 ((B A ∃!x A φ) → (ψ ↔ (x A φ) = B))
104, 9mpbiri 157 . . 3 ((B A ∃!x A φ) → ψ)
113, 10mpancom 399 . 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 1242  Ⅎwnf 1346   ∈ wcel 1390  ∃!wreu 2302  ℩crio 5410 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411 This theorem is referenced by: (None)
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