Theorem rspcimdv 2831

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
rspcimdv	|- ( 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 rspcimdv

Step	Hyp	Ref	Expression
1		df-ral 2449	. 2 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
2		rspcimdv.1	. . 3 |- ( ph -> A e. B )
3		simpr 109	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
4	3	eleq1d 2235	. . . . . 6 |- ( ( ph /\ x = A ) -> ( x e. B <-> A e. B ) )
5	4	biimprd 157	. . . . 5 |- ( ( ph /\ x = A ) -> ( A e. B -> x e. B ) )
6		rspcimdv.2	. . . . 5 |- ( ( ph /\ x = A ) -> ( ps -> ch ) )
7	5, 6	imim12d 74	. . . 4 |- ( ( ph /\ x = A ) -> ( ( x e. B -> ps ) -> ( A e. B -> ch ) ) )
8	2, 7	spcimdv 2810	. . 3 |- ( ph -> ( A. x ( x e. B -> ps ) -> ( A e. B -> ch ) ) )
9	2, 8	mpid 42	. 2 |- ( ph -> ( A. x ( x e. B -> ps ) -> ch ) )
10	1, 9	syl5bi 151	1 |- ( ph -> ( A. x e. B ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   = wceq 1343   e. wcel 2136   A.wral 2444

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728

This theorem is referenced by:  rspcdv  2833