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Theorem iota2df 4818
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 2033 . . 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 4808 . . 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 4796 . . . . 5 x(℩xψ)
1312a1i 9 . . . 4 (φx(℩xψ))
1413, 10nfeqd 2174 . . 3 (φ → Ⅎx(℩xψ) = B)
1511, 14nfbid 1462 . 2 (φ → Ⅎx(χ ↔ (℩xψ) = B))
161, 5, 8, 9, 10, 15vtocldf 2582 1 (φ → (χ ↔ (℩xψ) = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wnf 1329   wcel 1374  ∃!weu 1882  wnfc 2147  cio 4792
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794
This theorem is referenced by:  iota2d  4819  iota2  4820  riota2df  5412
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