Theorem vtoclef 1857

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

Hypotheses

Ref	Expression
vtoclef.1	|- (ph -> A.xph)
vtoclef.2	|- A e. V
vtoclef.3	|- (x = A -> ph)

Assertion

Ref	Expression
vtoclef	|- ph

Distinct variable group:   x,A

Proof of Theorem vtoclef

Step	Hyp	Ref	Expression
1		vtoclef.2	. . 3 |- A e. V
2	1	isseti 1815	. 2 |- E.x x = A
3		vtoclef.1	. . 3 |- (ph -> A.xph)
4		vtoclef.3	. . 3 |- (x = A -> ph)
5	3, 4	19.23ai 1064	. 2 |- (E.x x = A -> ph)
6	2, 5	ax-mp 7	1 |- ph

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  elabf 1896  nn0ind-raph 6214  cncnplem2 7775

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

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