Theorem rspc 2644

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Hypotheses

Ref	Expression
rspc.1	|- F/xps
rspc.2	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
rspc	|- (A e. B -> (A.x e. B ph -> ps))

Distinct variable groups:   x,A   x,B

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

Proof of Theorem rspc

Step	Hyp	Ref	Expression
1		df-ral 2305	. 2 |- (A.x e. B ph <-> A.x(x e. B -> ph))
2		nfcv 2175	. . . 4 |- F/_xA
3		nfv 1418	. . . . 5 |- F/x A e. B
4		rspc.1	. . . . 5 |- F/xps
5	3, 4	nfim 1461	. . . 4 |- F/x(A e. B -> ps)
6		eleq1 2097	. . . . 5 |- (x = A -> (x e. B <-> A e. B))
7		rspc.2	. . . . 5 |- (x = A -> (ph <-> ps))
8	6, 7	imbi12d 223	. . . 4 |- (x = A -> ((x e. B -> ph) <-> (A e. B -> ps)))
9	2, 5, 8	spcgf 2629	. . 3 |- (A e. B -> (A.x(x e. B -> ph) -> (A e. B -> ps)))
10	9	pm2.43a 45	. 2 |- (A e. B -> (A.x(x e. B -> ph) -> ps))
11	1, 10	syl5bi 141	1 |- (A e. B -> (A.x e. B ph -> ps))

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   = wceq 1242   F/wnf 1346   e. wcel 1390  A.wral 2300

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-ral 2305  df-v 2553

This theorem is referenced by:  rspcv  2646  rspc2  2655  pofun  4040  fmptcof  5274  fliftfuns  5381  qliftfuns  6126  bj-nntrans  9411