Theorem ceqsal 2884

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/xps
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/xps
3		ceqsal.3	. . 3 |- (x = A -> (ph <-> ps))
4	2, 3	ceqsalg 2883	. 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 176  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710  _Vcvv 2859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by:  ceqsalv  2885