Theorem spcimdv 2631

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.xps -> 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 1751	. 2 |- (ph -> A.x(x = A -> (ps -> ch)))
4		spcimdv.1	. 2 |- (ph -> A e. B)
5		nfv 1418	. . 3 |- F/xch
6		nfcv 2175	. . 3 |- F/_xA
7	5, 6	spcimgft 2623	. 2 |- (A.x(x = A -> (ps -> ch)) -> (A e. B -> (A.xps -> ch)))
8	3, 4, 7	sylc 56	1 |- (ph -> (A.xps -> ch))

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240   = wceq 1242   e. wcel 1390

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553

This theorem is referenced by:  spcdv  2632  rspcimdv  2651