Theorem vtocl 2827

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 1551	. 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 2826	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   _Vcvv 2772

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  vtoclb  2830  zfauscl  4165  bnd2  4218  uniex  4485  ordtriexmid  4570  onsucsssucexmid  4576  regexmid  4584  ordsoexmid  4611  onintexmid  4622  reg3exmid  4629  nnregexmid  4670  acexmidlemv  5944  caovcan  6113  findcard2  6988  findcard2s  6989  inffiexmid  7005  sup3exmid  9032  bj-uniex  15890  bj-omtrans  15929