Theorem rspcimdv 2790

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 2421	. 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 2208	. . . . . 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 2770	. . 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 1329   = wceq 1331   e. wcel 1480   A.wral 2416

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688

This theorem is referenced by:  rspcdv  2792