Theorem vtocl 1842

Description: Implicit substitution of a class for a set variable.

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

Proof of Theorem vtocl

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (ps -> A.xps)
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 1841	1 |- ps

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811

This theorem is referenced by:  vtoclb 1845  zfauscl 2705  pwex 2745  uniex 2870  fnbrfvb 3753  caoprcan 4055  zfregcl 4595  bnd2 4724  ac4c 4751  ac5 4752  kmlem2 4766  dominf 4904

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

Copyright terms: Public domain