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Theorem riota2df 5431
Description: A deduction version of riota2f 5432. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota2df.1 xφ
riota2df.2 (φxB)
riota2df.3 (φ → Ⅎxχ)
riota2df.4 (φB A)
riota2df.5 ((φ x = B) → (ψχ))
Assertion
Ref Expression
riota2df ((φ ∃!x A ψ) → (χ ↔ (x A ψ) = B))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   χ(x)   B(x)

Proof of Theorem riota2df
StepHypRef Expression
1 riota2df.4 . . . 4 (φB A)
21adantr 261 . . 3 ((φ ∃!x A ψ) → B A)
3 simpr 103 . . . 4 ((φ ∃!x A ψ) → ∃!x A ψ)
4 df-reu 2307 . . . 4 (∃!x A ψ∃!x(x A ψ))
53, 4sylib 127 . . 3 ((φ ∃!x A ψ) → ∃!x(x A ψ))
6 simpr 103 . . . . . 6 (((φ ∃!x A ψ) x = B) → x = B)
72adantr 261 . . . . . 6 (((φ ∃!x A ψ) x = B) → B A)
86, 7eqeltrd 2111 . . . . 5 (((φ ∃!x A ψ) x = B) → x A)
98biantrurd 289 . . . 4 (((φ ∃!x A ψ) x = B) → (ψ ↔ (x A ψ)))
10 riota2df.5 . . . . 5 ((φ x = B) → (ψχ))
1110adantlr 446 . . . 4 (((φ ∃!x A ψ) x = B) → (ψχ))
129, 11bitr3d 179 . . 3 (((φ ∃!x A ψ) x = B) → ((x A ψ) ↔ χ))
13 riota2df.1 . . . 4 xφ
14 nfreu1 2475 . . . 4 x∃!x A ψ
1513, 14nfan 1454 . . 3 x(φ ∃!x A ψ)
16 riota2df.3 . . . 4 (φ → Ⅎxχ)
1716adantr 261 . . 3 ((φ ∃!x A ψ) → Ⅎxχ)
18 riota2df.2 . . . 4 (φxB)
1918adantr 261 . . 3 ((φ ∃!x A ψ) → xB)
202, 5, 12, 15, 17, 19iota2df 4834 . 2 ((φ ∃!x A ψ) → (χ ↔ (℩x(x A ψ)) = B))
21 df-riota 5411 . . 3 (x A ψ) = (℩x(x A ψ))
2221eqeq1i 2044 . 2 ((x A ψ) = B ↔ (℩x(x A ψ)) = B)
2320, 22syl6bbr 187 1 ((φ ∃!x A ψ) → (χ ↔ (x A ψ) = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346   wcel 1390  ∃!weu 1897  wnfc 2162  ∃!wreu 2302  cio 4808  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-bnd 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-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411
This theorem is referenced by:  riota2f  5432  riota5f  5435
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