Theorem rspcdv 2914

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
rspcdv.1	|- ( ph -> A e. B )
rspcdv.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rspcdv	|- ( ph -> ( A. x e. B ps -> ch ) )

Distinct variable groups:   x, A   x, B   ph, x   ch, x

Allowed substitution hint:   ps( x)

Proof of Theorem rspcdv

Step	Hyp	Ref	Expression
1		rspcdv.1	. 2 |- ( ph -> A e. B )
2		rspcdv.2	. . 3 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3	2	biimpd 144	. 2 |- ( ( ph /\ x = A ) -> ( ps -> ch ) )
4	1, 3	rspcimdv 2912	1 |- ( ph -> ( A. x e. B ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> 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:  ralxfrd  4565  suppofss1dcl  6442  suppofss2dcl  6443  prloc  7771  zindd  9659  wrd2ind  11370  mpomulcn  15377  fsumdvdsmul  15805  uspgr2wlkeq  16306  umgr2cwwk2dif  16365