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Theorem iota2d 6019
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 1995 . 2 𝑥𝜑
5 nfvd 1996 . 2 (𝜑 → Ⅎ𝑥𝜒)
6 nfcvd 2914 . 2 (𝜑𝑥𝐵)
71, 2, 3, 4, 5, 6iota2df 6018 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  ∃!weu 2618  cio 5992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-sbc 3588  df-un 3728  df-sn 4317  df-pr 4319  df-uni 4575  df-iota 5994
This theorem is referenced by:  erov  7997  psgnvalii  18136  q1peqb  24134
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