Theorem vtocl 2832

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 1552	. 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 2831	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   _Vcvv 2776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778

This theorem is referenced by:  vtoclb  2835  zfauscl  4180  bnd2  4233  uniex  4502  ordtriexmid  4587  onsucsssucexmid  4593  regexmid  4601  ordsoexmid  4628  onintexmid  4639  reg3exmid  4646  nnregexmid  4687  acexmidlemv  5965  caovcan  6134  findcard2  7012  findcard2s  7013  inffiexmid  7029  sup3exmid  9065  bj-uniex  16052  bj-omtrans  16091