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Theorem vtoclgf 2671
Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtoclgf.1 𝑥𝐴
vtoclgf.2 𝑥𝜓
vtoclgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclgf.4 𝜑
Assertion
Ref Expression
vtoclgf (𝐴𝑉𝜓)

Proof of Theorem vtoclgf
StepHypRef Expression
1 elex 2624 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtoclgf.1 . . . 4 𝑥𝐴
32issetf 2620 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 vtoclgf.2 . . . 4 𝑥𝜓
5 vtoclgf.4 . . . . 5 𝜑
6 vtoclgf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6mpbii 146 . . . 4 (𝑥 = 𝐴𝜓)
84, 7exlimi 1528 . . 3 (∃𝑥 𝑥 = 𝐴𝜓)
93, 8sylbi 119 . 2 (𝐴 ∈ V → 𝜓)
101, 9syl 14 1 (𝐴𝑉𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wnf 1392  wex 1424  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:  vtoclg  2672  vtocl2gf  2674  vtocl3gf  2675  vtoclgaf  2677  ceqsexg  2736  elabgf  2749  mob  2788  opeliunxp2  4543  fvmptss2  5336  isummolem2a  10653
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