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Theorem iota2df 4365
 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.6 . 2 (φxB)
2 iota2df.5 . . 3 (φ → Ⅎxχ)
3 nfiota1 4341 . . . . 5 x(℩xψ)
43a1i 10 . . . 4 (φx(℩xψ))
54, 1nfeqd 2503 . . 3 (φ → Ⅎx(℩xψ) = B)
62, 5nfbid 1832 . 2 (φ → Ⅎx(χ ↔ (℩xψ) = B))
7 iota2df.4 . . 3 xφ
8 iota2df.3 . . . . 5 ((φ x = B) → (ψχ))
9 simpr 447 . . . . . 6 ((φ x = B) → x = B)
109eqeq2d 2364 . . . . 5 ((φ x = B) → ((℩xψ) = x ↔ (℩xψ) = B))
118, 10bibi12d 312 . . . 4 ((φ x = B) → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B)))
1211ex 423 . . 3 (φ → (x = B → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B))))
137, 12alrimi 1765 . 2 (φx(x = B → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B))))
14 iota2df.2 . . . 4 (φ∃!xψ)
15 iota1 4353 . . . 4 (∃!xψ → (ψ ↔ (℩xψ) = x))
1614, 15syl 15 . . 3 (φ → (ψ ↔ (℩xψ) = x))
177, 16alrimi 1765 . 2 (φx(ψ ↔ (℩xψ) = x))
18 iota2df.1 . 2 (φB V)
19 vtoclgft 2905 . 2 (((xB x(χ ↔ (℩xψ) = B)) (x(x = B → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B))) x(ψ ↔ (℩xψ) = x)) B V) → (χ ↔ (℩xψ) = B))
201, 6, 13, 17, 18, 19syl221anc 1193 1 (φ → (χ ↔ (℩xψ) = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Ⅎwnfc 2476  ℩cio 4337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  iota2d  4366  iota2  4367
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