Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtoclgf Structured version   GIF version

Theorem vtoclgf 2589
 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 2543 . 2 (A 𝑉A V)
2 vtoclgf.1 . . . 4 xA
32issetf 2540 . . 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 1467 . . 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 1228  Ⅎwnf 1329  ∃wex 1362   ∈ wcel 1374  Ⅎwnfc 2147  Vcvv 2535 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537 This theorem is referenced by:  vtoclg  2590  vtocl2gf  2592  vtocl3gf  2593  vtoclgaf  2595  ceqsexg  2649  elabgf  2662  mob  2700  opeliunxp2  4403  fvmptss2  5172
 Copyright terms: Public domain W3C validator