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Theorem iota2df 4834
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))
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 (φB 𝑉)
2 iota2df.3 . . 3 ((φ x = B) → (ψχ))
3 simpr 103 . . . 4 ((φ x = B) → x = B)
43eqeq2d 2048 . . 3 ((φ x = B) → ((℩xψ) = x ↔ (℩xψ) = B))
52, 4bibi12d 224 . 2 ((φ x = B) → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B)))
6 iota2df.2 . . 3 (φ∃!xψ)
7 iota1 4824 . . 3 (∃!xψ → (ψ ↔ (℩xψ) = x))
86, 7syl 14 . 2 (φ → (ψ ↔ (℩xψ) = x))
9 iota2df.4 . 2 xφ
10 iota2df.6 . 2 (φxB)
11 iota2df.5 . . 3 (φ → Ⅎxχ)
12 nfiota1 4812 . . . . 5 x(℩xψ)
1312a1i 9 . . . 4 (φx(℩xψ))
1413, 10nfeqd 2189 . . 3 (φ → Ⅎx(℩xψ) = B)
1511, 14nfbid 1477 . 2 (φ → Ⅎx(χ ↔ (℩xψ) = B))
161, 5, 8, 9, 10, 15vtocldf 2599 1 (φ → (χ ↔ (℩xψ) = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346   wcel 1390  ∃!weu 1897  wnfc 2162  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-bndl 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:  iota2d  4835  iota2  4836  riota2df  5431
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