New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iota2d GIF version

Theorem iota2d 4366
 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 (φB V)
iota2df.2 (φ∃!xψ)
iota2df.3 ((φ x = 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 (φB V)
2 iota2df.2 . 2 (φ∃!xψ)
3 iota2df.3 . 2 ((φ x = B) → (ψχ))
4 nfv 1619 . 2 xφ
5 nfvd 1620 . 2 (φ → Ⅎxχ)
6 nfcvd 2490 . 2 (φxB)
71, 2, 3, 4, 5, 6iota2df 4365 1 (φ → (χ ↔ (℩xψ) = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator