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Theorem iota2df 5834
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 477 . . . 4 ((𝜑𝑥 = 𝐵) → 𝑥 = 𝐵)
43eqeq2d 2631 . . 3 ((𝜑𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
52, 4bibi12d 335 . 2 ((𝜑𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6 iota2df.2 . . 3 (𝜑 → ∃!𝑥𝜓)
7 iota1 5824 . . 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 5812 . . . . 5 𝑥(℩𝑥𝜓)
1312a1i 11 . . . 4 (𝜑𝑥(℩𝑥𝜓))
1413, 10nfeqd 2768 . . 3 (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
1511, 14nfbid 1829 . 2 (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
161, 5, 8, 9, 10, 15vtocldf 3242 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wnf 1705  wcel 1987  ∃!weu 2469  wnfc 2748  cio 5808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-sbc 3418  df-un 3560  df-sn 4149  df-pr 4151  df-uni 4403  df-iota 5810
This theorem is referenced by:  iota2d  5835  iota2  5836  riota2df  6585  opiota  7174
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