Theorem ceqsal 2755

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

Assertion

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

Distinct variable group:   x, A

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

Proof of Theorem ceqsal

Step	Hyp	Ref	Expression
1		ceqsal.2	. 2 |- A e. _V
2		ceqsal.1	. . 3 |- F/ x ps
3		ceqsal.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	2, 3	ceqsalg 2754	. 2 |- ( A e. _V -> ( A. x ( x = A -> ph ) <-> ps ) )
5	1, 4	ax-mp 5	1 |- ( A. x ( x = A -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   F/wnf 1448   e. wcel 2136   _Vcvv 2726

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  ceqsalv  2756