Theorem rspcedv 2705

Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-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
rspcedv	|- ( ph -> ( ch -> E. x e. B ps ) )

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

Allowed substitution hint:   ps( x)

Proof of Theorem rspcedv

Step	Hyp	Ref	Expression
1		rspcdv.1	. 2 |- ( ph -> A e. B )
2		rspcdv.2	. . 3 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3	2	biimprd 156	. 2 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
4	1, 3	rspcimedv 2703	1 |- ( ph -> ( ch -> E. x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603

This theorem is referenced by:  rspcedvd  2708  rexxfrd  4213  ltexnqq  6598  halfnqq  6600  ltbtwnnqq  6605  genpml  6707  genpmu  6708  genprndl  6711  genprndu  6712  axarch  7057  apreap  7687