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Theorem iota2d 5113
 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))
Assertion
Ref Expression
iota2d (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem iota2d
StepHypRef Expression
1 iota2df.1 . 2 (𝜑𝐵𝑉)
2 iota2df.2 . 2 (𝜑 → ∃!𝑥𝜓)
3 iota2df.3 . 2 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
4 nfv 1508 . 2 𝑥𝜑
5 nfvd 1509 . 2 (𝜑 → Ⅎ𝑥𝜒)
6 nfcvd 2282 . 2 (𝜑𝑥𝐵)
71, 2, 3, 4, 5, 6iota2df 5112 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∃!weu 1999  ℩cio 5086 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088 This theorem is referenced by: (None)
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