Theorem ceqsalv 3098

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

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

Allowed substitution hint:   ph( x)

Proof of Theorem ceqsalv

Step	Hyp	Ref	Expression
1		nfv 1674	. 2 |- F/ x ps
2		ceqsalv.1	. 2 |- A e. _V
3		ceqsalv.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	ceqsal 3097	1 |- ( A. x ( x = A -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1368   = wceq 1370   e. wcel 1758   _Vcvv 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3072

This theorem is referenced by:  ralxpxfr2d  3183  clel2  3195  clel4  3198  reu8  3254  frsn  5009  raliunxp  5079  fv3  5804  funimass4  5843  marypha2lem3  7790  kmlem12  8433  fpwwe2lem12  8911  vdwmc2  14144  itg2leub  21330  nmoubi  24309  choc0  24866  nmopub  25449  nmfnleub  25466  heibor1lem  28848