Theorem vtoclri 2882

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)

Hypotheses

Ref	Expression
vtoclri.1	|- ( x = A -> ( ph <-> ps ) )
vtoclri.2	|- A. x e. B ph

Assertion

Ref	Expression
vtoclri	|- ( A e. B -> ps )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem vtoclri

Step	Hyp	Ref	Expression
1		vtoclri.1	. 2 |- ( x = A -> ( ph <-> ps ) )
2		vtoclri.2	. . 3 |- A. x e. B ph
3	2	rspec 2585	. 2 |- ( x e. B -> ph )
4	1, 3	vtoclga 2871	1 |- ( A e. B -> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805

This theorem is referenced by:  ordpwsucexmid  4674  bj-nn0suc0  16666