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Theorem riota2df 5827
Description: A deduction version of riota2f 5828. (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 274 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐵𝐴)
3 simpr 109 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 𝜓)
4 df-reu 2455 . . . 4 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
53, 4sylib 121 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥(𝑥𝐴𝜓))
6 simpr 109 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
72adantr 274 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵𝐴)
86, 7eqeltrd 2247 . . . . 5 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥𝐴)
98biantrurd 303 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥𝐴𝜓)))
10 riota2df.5 . . . . 5 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
1110adantlr 474 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓𝜒))
129, 11bitr3d 189 . . 3 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥𝐴𝜓) ↔ 𝜒))
13 riota2df.1 . . . 4 𝑥𝜑
14 nfreu1 2641 . . . 4 𝑥∃!𝑥𝐴 𝜓
1513, 14nfan 1558 . . 3 𝑥(𝜑 ∧ ∃!𝑥𝐴 𝜓)
16 riota2df.3 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
1716adantr 274 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → Ⅎ𝑥𝜒)
18 riota2df.2 . . . 4 (𝜑𝑥𝐵)
1918adantr 274 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝑥𝐵)
202, 5, 12, 15, 17, 19iota2df 5182 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵))
21 df-riota 5807 . . 3 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
2221eqeq1i 2178 . 2 ((𝑥𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵)
2320, 22bitr4di 197 1 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wnf 1453  ∃!weu 2019  wcel 2141  wnfc 2299  ∃!wreu 2450  cio 5156  crio 5806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3587  df-pr 3588  df-uni 3795  df-iota 5158  df-riota 5807
This theorem is referenced by:  riota2f  5828  riota5f  5831
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