Theorem ceqsal 2792

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 2791	. 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 105   A.wal 1362   = wceq 1364   F/wnf 1474   e. wcel 2167   _Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  ceqsalv  2793