Theorem ceqsalv 2760

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 1521	. 2 |- F/ x ps
2		ceqsalv.1	. 2 |- A e. _V
3		ceqsalv.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	ceqsal 2759	1 |- ( A. x ( x = A -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   _Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  gencbval  2778  clel2  2863  clel4  2866  reu8  2926  raliunxp  4752  fv3  5519