Theorem rspcimdv 2957

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 2620	. 2 |- (A.x e. B ps <-> A.x(x e. B -> ps))
2		rspcimdv.1	. . 3 |- (ph -> A e. B)
3		simpr 447	. . . . . . 7 |- ((ph /\ x = A) -> x = A)
4	3	eleq1d 2419	. . . . . 6 |- ((ph /\ x = A) -> (x e. B <-> A e. B))
5	4	biimprd 214	. . . . 5 |- ((ph /\ x = A) -> (A e. B -> x e. B))
6		rspcimdv.2	. . . . 5 |- ((ph /\ x = A) -> (ps -> ch))
7	5, 6	imim12d 68	. . . 4 |- ((ph /\ x = A) -> ((x e. B -> ps) -> (A e. B -> ch)))
8	2, 7	spcimdv 2937	. . 3 |- (ph -> (A.x(x e. B -> ps) -> (A e. B -> ch)))
9	2, 8	mpid 37	. 2 |- (ph -> (A.x(x e. B -> ps) -> ch))
10	1, 9	syl5bi 208	1 |- (ph -> (A.x e. B ps -> ch))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862

This theorem is referenced by:  rspcimedv  2958  rspcdv  2959