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Theorem iota2d 5005
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 1466 . 2 𝑥𝜑
5 nfvd 1467 . 2 (𝜑 → Ⅎ𝑥𝜒)
6 nfcvd 2229 . 2 (𝜑𝑥𝐵)
71, 2, 3, 4, 5, 6iota2df 5004 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  ∃!weu 1948  cio 4978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-sn 3452  df-pr 3453  df-uni 3654  df-iota 4980
This theorem is referenced by: (None)
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