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Theorem iota2d 6421
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 1917 . 2 𝑥𝜑
5 nfvd 1918 . 2 (𝜑 → Ⅎ𝑥𝜒)
6 nfcvd 2908 . 2 (𝜑𝑥𝐵)
71, 2, 3, 4, 5, 6iota2df 6420 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  ∃!weu 2568  cio 6389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391
This theorem is referenced by:  erov  8603  psgnvalii  19117  q1peqb  25319
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