MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iota2df Structured version   Visualization version   GIF version

Theorem iota2df 6017
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 471 . . . 4 ((𝜑𝑥 = 𝐵) → 𝑥 = 𝐵)
43eqeq2d 2781 . . 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 6007 . . 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 5995 . . . . 5 𝑥(℩𝑥𝜓)
1312a1i 11 . . . 4 (𝜑𝑥(℩𝑥𝜓))
1413, 10nfeqd 2921 . . 3 (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
1511, 14nfbid 1984 . 2 (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
161, 5, 8, 9, 10, 15vtocldf 3407 1 (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wnf 1856  wcel 2145  ∃!weu 2618  wnfc 2900  cio 5991
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 837  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 4318  df-pr 4320  df-uni 4576  df-iota 5993
This theorem is referenced by:  iota2d  6018  iota2  6019  riota2df  6777  opiota  7382
  Copyright terms: Public domain W3C validator