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Theorem vtocl2gf 2674
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
Hypotheses
Ref Expression
vtocl2gf.1 𝑥𝐴
vtocl2gf.2 𝑦𝐴
vtocl2gf.3 𝑦𝐵
vtocl2gf.4 𝑥𝜓
vtocl2gf.5 𝑦𝜒
vtocl2gf.6 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2gf.7 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2gf.8 𝜑
Assertion
Ref Expression
vtocl2gf ((𝐴𝑉𝐵𝑊) → 𝜒)

Proof of Theorem vtocl2gf
StepHypRef Expression
1 elex 2624 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtocl2gf.3 . . 3 𝑦𝐵
3 vtocl2gf.2 . . . . 5 𝑦𝐴
43nfel1 2235 . . . 4 𝑦 𝐴 ∈ V
5 vtocl2gf.5 . . . 4 𝑦𝜒
64, 5nfim 1507 . . 3 𝑦(𝐴 ∈ V → 𝜒)
7 vtocl2gf.7 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
87imbi2d 228 . . 3 (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
9 vtocl2gf.1 . . . 4 𝑥𝐴
10 vtocl2gf.4 . . . 4 𝑥𝜓
11 vtocl2gf.6 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
12 vtocl2gf.8 . . . 4 𝜑
139, 10, 11, 12vtoclgf 2671 . . 3 (𝐴 ∈ V → 𝜓)
142, 6, 8, 13vtoclgf 2671 . 2 (𝐵𝑊 → (𝐴 ∈ V → 𝜒))
151, 14mpan9 275 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wnf 1392  wcel 1436  wnfc 2212  Vcvv 2615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  vtocl3gf  2675  vtocl2g  2676  vtocl2gaf  2679
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