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Theorem iota2df 5792
 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))
iota2df.4 𝑥𝜑
iota2df.5 (𝜑 → Ⅎ𝑥𝜒)
iota2df.6 (𝜑𝑥𝐵)
Assertion
Ref Expression
iota2df (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Proof of Theorem iota2df
StepHypRef Expression
1 iota2df.1 . 2 (𝜑𝐵𝑉)
2 iota2df.3 . . 3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
3 simpr 476 . . . 4 ((𝜑𝑥 = 𝐵) → 𝑥 = 𝐵)
43eqeq2d 2620 . . 3 ((𝜑𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
52, 4bibi12d 334 . 2 ((𝜑𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6 iota2df.2 . . 3 (𝜑 → ∃!𝑥𝜓)
7 iota1 5782 . . 3 (∃!𝑥𝜓 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥))
86, 7syl 17 . 2 (𝜑 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥))
9 iota2df.4 . 2 𝑥𝜑
10 iota2df.6 . 2 (𝜑𝑥𝐵)
11 iota2df.5 . . 3 (𝜑 → Ⅎ𝑥𝜒)
12 nfiota1 5770 . . . . 5 𝑥(℩𝑥𝜓)
1312a1i 11 . . . 4 (𝜑𝑥(℩𝑥𝜓))
1413, 10nfeqd 2758 . . 3 (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
1511, 14nfbid 1820 . 2 (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
161, 5, 8, 9, 10, 15vtocldf 3229 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∃!weu 2458  Ⅎwnfc 2738  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768 This theorem is referenced by:  iota2d  5793  iota2  5794  riota2df  6531  opiota  7118
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