Theorem rspcimdv 2878

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 2489	. 2 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
2		rspcimdv.1	. . 3 |- ( ph -> A e. B )
3		simpr 110	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
4	3	eleq1d 2274	. . . . . 6 |- ( ( ph /\ x = A ) -> ( x e. B <-> A e. B ) )
5	4	biimprd 158	. . . . 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 2857	. . 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	biimtrid 152	1 |- ( ph -> ( A. x e. B ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   e. wcel 2176   A.wral 2484

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774

This theorem is referenced by:  rspcdv  2880  mpomulcn  15071