Theorem vtoclefex 33181

Description: Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)

Hypotheses

Ref	Expression
vtoclefex.1	|- F/ x ph
vtoclefex.3	|- ( x = A -> ph )

Assertion

Ref	Expression
vtoclefex	|- ( A e. V -> ph )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem vtoclefex

Step	Hyp	Ref	Expression
1		vtoclefex.1	. 2 |- F/ x ph
2		vtoclefex.3	. . 3 |- ( x = A -> ph )
3	2	ax-gen 1722	. 2 |- A. x ( x = A -> ph )
4		vtoclegft 3280	. 2 |- ( ( A e. V /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph )
5	1, 3, 4	mp3an23 1416	1 |- ( A e. V -> ph )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  finxpreclem2  33227