Theorem spcimdv 2637

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

Hypotheses

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

Assertion

Ref	Expression
spcimdv	|- ( ph -> ( A. x ps -> ch ) )

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

Allowed substitution hints:   ps( x)   B( x)

Proof of Theorem spcimdv

Step	Hyp	Ref	Expression
1		spcimdv.2	. . . 4 |- ( ( ph /\ x = A ) -> ( ps -> ch ) )
2	1	ex 108	. . 3 |- ( ph -> ( x = A -> ( ps -> ch ) ) )
3	2	alrimiv 1754	. 2 |- ( ph -> A. x ( x = A -> ( ps -> ch ) ) )
4		spcimdv.1	. 2 |- ( ph -> A e. B )
5		nfv 1421	. . 3 |- F/ x ch
6		nfcv 2178	. . 3 |- F/_ x A
7	5, 6	spcimgft 2629	. 2 |- ( A. x ( x = A -> ( ps -> ch ) ) -> ( A e. B -> ( A. x ps -> ch ) ) )
8	3, 4, 7	sylc 56	1 |- ( ph -> ( A. x ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  spcdv  2638  rspcimdv  2657