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Theorem vtocl2gf 2752
 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 2701 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtocl2gf.3 . . 3 𝑦𝐵
3 vtocl2gf.2 . . . . 5 𝑦𝐴
43nfel1 2293 . . . 4 𝑦 𝐴 ∈ V
5 vtocl2gf.5 . . . 4 𝑦𝜒
64, 5nfim 1552 . . 3 𝑦(𝐴 ∈ V → 𝜒)
7 vtocl2gf.7 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
87imbi2d 229 . . 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 2748 . . 3 (𝐴 ∈ V → 𝜓)
142, 6, 8, 13vtoclgf 2748 . 2 (𝐵𝑊 → (𝐴 ∈ V → 𝜒))
151, 14mpan9 279 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  Ⅎwnf 1437   ∈ wcel 1481  Ⅎwnfc 2269  Vcvv 2690 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692 This theorem is referenced by:  vtocl3gf  2753  vtocl2g  2754  vtocl2gaf  2757
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