Theorem vtocl 2792

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 2791	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2738

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 2740

This theorem is referenced by:  vtoclb  2795  zfauscl  4124  bnd2  4174  uniex  4438  ordtriexmid  4521  onsucsssucexmid  4527  regexmid  4535  ordsoexmid  4562  onintexmid  4573  reg3exmid  4580  nnregexmid  4621  acexmidlemv  5873  caovcan  6039  findcard2  6889  findcard2s  6890  inffiexmid  6906  sup3exmid  8914  bj-uniex  14672  bj-omtrans  14711