Theorem ceqsal 2583

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 2582	. 2 |- ( A e. _V -> ( A. x ( x = A -> ph ) <-> ps ) )
5	1, 4	ax-mp 7	1 |- ( A. x ( x = A -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   F/wnf 1349   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  ceqsalv  2584