Theorem vtocl 1839

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-17 970	. 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 1838	1 |- ps

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  Vcvv 1808

This theorem is referenced by:  vtoclb 1842  zfauscl 2701  pwex 2741  uniex 2866  fnbrfvb 3748  caoprcan 4050  zfregcl 4578  bnd2 4707  ac4c 4734  ac5 4735  kmlem2 4749  dominf 4887

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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