Theorem rspcdva 2707

Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Hypotheses

Ref	Expression
rspcdva.1	|- ( x = C -> ( ps <-> ch ) )
rspcdva.2	|- ( ph -> A. x e. A ps )
rspcdva.3	|- ( ph -> C e. A )

Assertion

Ref	Expression
rspcdva	|- ( ph -> ch )

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

Allowed substitution hint:   ps( x)

Proof of Theorem rspcdva

Step	Hyp	Ref	Expression
1		rspcdva.2	. 2 |- ( ph -> A. x e. A ps )
2		rspcdva.3	. . 3 |- ( ph -> C e. A )
3		rspcdva.1	. . . 4 |- ( x = C -> ( ps <-> ch ) )
4	3	adantl 271	. . 3 |- ( ( ph /\ x = C ) -> ( ps <-> ch ) )
5	2, 4	rspcdv 2704	. 2 |- ( ph -> ( A. x e. A ps -> ch ) )
6	1, 5	mpd 13	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348

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-ral 2353  df-v 2603

This theorem is referenced by:  uzsinds  9428  iseqz  9469  bezoutlemex  10390  bezoutlemzz  10391  bezoutlemmo  10395  bezoutlemle  10397  bezoutlemsup  10398