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Theorem iota2d 4835
 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 1418 . 2 xφ
5 nfvd 1419 . 2 (φ → Ⅎxχ)
6 nfcvd 2176 . 2 (φxB)
71, 2, 3, 4, 5, 6iota2df 4834 1 (φ → (χ ↔ (℩xψ) = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  ℩cio 4808 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810 This theorem is referenced by: (None)
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