Theorem vtocl 3130

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)

Hypotheses

Ref	Expression
vtocl.1	|- A e. _V
vtocl.2	|- ( x = A -> ( ph <-> ps ) )
vtocl.3	|- ph

Assertion

Ref	Expression
vtocl	|- ps

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem vtocl

Step	Hyp	Ref	Expression
1		nfv 1674	. 2 |- F/ x ps
2		vtocl.1	. 2 |- A e. _V
3		vtocl.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4		vtocl.3	. 2 |- ph
5	1, 2, 3, 4	vtoclf 3129	1 |- ps

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1370   e. wcel 1758   _Vcvv 3078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3080

This theorem is referenced by:  vtoclb  3133  zfauscl  4526  pwex  4586  fnbrfvb  5844  caovcan  6380  uniex  6489  findcard2  7666  zfregcl  7923  bnd2  8214  kmlem2  8434  axcc2lem  8719  dominf  8728  dcomex  8730  ac4c  8759  ac5  8760  dominfac  8851  pwfseqlem4  8943  grothomex  9110  ramub2  14196  ismred2  14663  dvfsumlem2  21635  plydivlem4  21898  frmin  27867  voliunnfl  28603  volsupnfl  28604  prdsbnd2  28862  iscringd  28967  monotoddzzfi  29451  monotoddzz  29452  bnj865  32268  bnj1015  32306