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Theorem iota2d 5202
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 1528 . 2 𝑥𝜑
5 nfvd 1529 . 2 (𝜑 → Ⅎ𝑥𝜒)
6 nfcvd 2320 . 2 (𝜑𝑥𝐵)
71, 2, 3, 4, 5, 6iota2df 5201 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  ∃!weu 2026  wcel 2148  cio 5175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3598  df-pr 3599  df-uni 3810  df-iota 5177
This theorem is referenced by: (None)
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