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Theorem iota2df 4891
 Description: A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
Hypotheses
Ref Expression
iota2df.1 (𝜑𝐵𝑉)
iota2df.2 (𝜑 → ∃!𝑥𝜓)
iota2df.3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
iota2df.4 𝑥𝜑
iota2df.5 (𝜑 → Ⅎ𝑥𝜒)
iota2df.6 (𝜑𝑥𝐵)
Assertion
Ref Expression
iota2df (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Proof of Theorem iota2df
StepHypRef Expression
1 iota2df.1 . 2 (𝜑𝐵𝑉)
2 iota2df.3 . . 3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
3 simpr 103 . . . 4 ((𝜑𝑥 = 𝐵) → 𝑥 = 𝐵)
43eqeq2d 2051 . . 3 ((𝜑𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
52, 4bibi12d 224 . 2 ((𝜑𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6 iota2df.2 . . 3 (𝜑 → ∃!𝑥𝜓)
7 iota1 4881 . . 3 (∃!𝑥𝜓 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥))
86, 7syl 14 . 2 (𝜑 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥))
9 iota2df.4 . 2 𝑥𝜑
10 iota2df.6 . 2 (𝜑𝑥𝐵)
11 iota2df.5 . . 3 (𝜑 → Ⅎ𝑥𝜒)
12 nfiota1 4869 . . . . 5 𝑥(℩𝑥𝜓)
1312a1i 9 . . . 4 (𝜑𝑥(℩𝑥𝜓))
1413, 10nfeqd 2192 . . 3 (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
1511, 14nfbid 1480 . 2 (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
161, 5, 8, 9, 10, 15vtocldf 2605 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  ∃!weu 1900  Ⅎwnfc 2165  ℩cio 4865 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867 This theorem is referenced by:  iota2d  4892  iota2  4893  riota2df  5488
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