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Theorem riota2df 5515
Description: A deduction version of riota2f 5516. (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 (𝜑𝐵𝐴)
riota2df.5 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
Assertion
Ref Expression
riota2df ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 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 (𝜑𝐵𝐴)
21adantr 265 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐵𝐴)
3 simpr 107 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 𝜓)
4 df-reu 2330 . . . 4 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
53, 4sylib 131 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥(𝑥𝐴𝜓))
6 simpr 107 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
72adantr 265 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵𝐴)
86, 7eqeltrd 2130 . . . . 5 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥𝐴)
98biantrurd 293 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥𝐴𝜓)))
10 riota2df.5 . . . . 5 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
1110adantlr 454 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓𝜒))
129, 11bitr3d 183 . . 3 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥𝐴𝜓) ↔ 𝜒))
13 riota2df.1 . . . 4 𝑥𝜑
14 nfreu1 2498 . . . 4 𝑥∃!𝑥𝐴 𝜓
1513, 14nfan 1473 . . 3 𝑥(𝜑 ∧ ∃!𝑥𝐴 𝜓)
16 riota2df.3 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
1716adantr 265 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → Ⅎ𝑥𝜒)
18 riota2df.2 . . . 4 (𝜑𝑥𝐵)
1918adantr 265 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝑥𝐵)
202, 5, 12, 15, 17, 19iota2df 4918 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵))
21 df-riota 5495 . . 3 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
2221eqeq1i 2063 . 2 ((𝑥𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵)
2320, 22syl6bbr 191 1 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wcel 1409  ∃!weu 1916  wnfc 2181  ∃!wreu 2325  cio 4892  crio 5494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-reu 2330  df-v 2576  df-sbc 2787  df-un 2949  df-sn 3408  df-pr 3409  df-uni 3608  df-iota 4894  df-riota 5495
This theorem is referenced by:  riota2f  5516  riota5f  5519
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