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Theorem vtoclgf 2606
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 2560 . 2 (A 𝑉A V)
2 vtoclgf.1 . . . 4 xA
32issetf 2556 . . 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 136 . . . 4 (x = Aψ)
84, 7exlimi 1482 . . 3 (x x = Aψ)
93, 8sylbi 114 . 2 (A V → ψ)
101, 9syl 14 1 (A 𝑉ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wnf 1346  wex 1378   wcel 1390  wnfc 2162  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  vtoclg  2607  vtocl2gf  2609  vtocl3gf  2610  vtoclgaf  2612  ceqsexg  2666  elabgf  2679  mob  2717  opeliunxp2  4419  fvmptss2  5190
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