Theorem vtocl 2791

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 1528	. 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 2790	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2737

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739

This theorem is referenced by:  vtoclb  2794  zfauscl  4123  bnd2  4173  uniex  4437  ordtriexmid  4520  onsucsssucexmid  4526  regexmid  4534  ordsoexmid  4561  onintexmid  4572  reg3exmid  4579  nnregexmid  4620  acexmidlemv  5872  caovcan  6038  findcard2  6888  findcard2s  6889  inffiexmid  6905  sup3exmid  8913  bj-uniex  14639  bj-omtrans  14678