Theorem vtoclri 2805

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 2522	. 2 |- ( x e. B -> ph )
4	1, 3	vtoclga 2796	1 |- ( A e. B -> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732

This theorem is referenced by:  ordpwsucexmid  4554  bj-nn0suc0  13985