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Theorem iota2df 5119
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 109 . . . 4 ((𝜑𝑥 = 𝐵) → 𝑥 = 𝐵)
43eqeq2d 2152 . . 3 ((𝜑𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
52, 4bibi12d 234 . 2 ((𝜑𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6 iota2df.2 . . 3 (𝜑 → ∃!𝑥𝜓)
7 iota1 5109 . . 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 5097 . . . . 5 𝑥(℩𝑥𝜓)
1312a1i 9 . . . 4 (𝜑𝑥(℩𝑥𝜓))
1413, 10nfeqd 2297 . . 3 (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
1511, 14nfbid 1568 . 2 (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
161, 5, 8, 9, 10, 15vtocldf 2740 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wnf 1437  wcel 1481  ∃!weu 2000  wnfc 2269  cio 5093
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-sn 3537  df-pr 3538  df-uni 3744  df-iota 5095
This theorem is referenced by:  iota2d  5120  iota2  5121  riota2df  5757
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