Theorem ceqsalv 1818

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis.

Hypotheses

Ref	Expression
ceqsalv.1	|- A e. V
ceqsalv.2	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
ceqsalv	|- (A.x(x = A -> ph) <-> ps)

Distinct variable groups:   x,A   ps,x

Proof of Theorem ceqsalv

Step	Hyp	Ref	Expression
1		ax-17 968	. 2 |- (ps -> A.xps)
2		ceqsalv.1	. 2 |- A e. V
3		ceqsalv.2	. 2 |- (x = A -> (ph <-> ps))
4	1, 2, 3	ceqsal 1817	1 |- (A.x(x = A -> ph) <-> ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   = wceq 953   e. wcel 955  Vcvv 1802

This theorem is referenced by:  clel2 1882  clel4 1885  reu8 1926  prsspw 2471  fv3 3718  funimass4 3748  ranksn 4661  kmlem12 4748  choc0 9205  h1deot 9387

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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