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Theorem vtoclgf 2913
 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 (A Vψ)

Proof of Theorem vtoclgf
StepHypRef Expression
1 elex 2867 . 2 (A VA V)
2 vtoclgf.1 . . . 4 xA
32issetf 2864 . . 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 202 . . . 4 (x = Aψ)
84, 7exlimi 1803 . . 3 (x x = Aψ)
93, 8sylbi 187 . 2 (A V → ψ)
101, 9syl 15 1 (A Vψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by:  vtoclg  2914  vtocl2gf  2916  vtocl3gf  2917  vtoclgaf  2919  ceqsexg  2970  elabgf  2983  mob  3018  opeliunxp2  4822
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