Theorem rspc3ev 2966

Description: 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)

Hypotheses

Ref	Expression
rspc3v.1	|- (x = A -> (ph <-> ch))
rspc3v.2	|- (y = B -> (ch <-> th))
rspc3v.3	|- (z = C -> (th <-> ps))

Assertion

Ref	Expression
rspc3ev	|- (((A e. R /\ B e. S /\ C e. T) /\ ps) -> E.x e. R E.y e. S E.z e. T ph)

Distinct variable groups:   ps,z   ch,x   th,y   x,y,z,A   y,B,z   z,C   x,R   x,S,y   x,T,y,z

Allowed substitution hints:   ph(x,y,z)   ps(x,y)   ch(y,z)   th(x,z)   B(x)   C(x,y)   R(y,z)   S(z)

Proof of Theorem rspc3ev

Step	Hyp	Ref	Expression
1		simpl1 958	. 2 |- (((A e. R /\ B e. S /\ C e. T) /\ ps) -> A e. R)
2		simpl2 959	. 2 |- (((A e. R /\ B e. S /\ C e. T) /\ ps) -> B e. S)
3		rspc3v.3	. . . 4 |- (z = C -> (th <-> ps))
4	3	rspcev 2956	. . 3 |- ((C e. T /\ ps) -> E.z e. T th)
5	4	3ad2antl3 1119	. 2 |- (((A e. R /\ B e. S /\ C e. T) /\ ps) -> E.z e. T th)
6		rspc3v.1	. . . 4 |- (x = A -> (ph <-> ch))
7	6	rexbidv 2636	. . 3 |- (x = A -> (E.z e. T ph <-> E.z e. T ch))
8		rspc3v.2	. . . 4 |- (y = B -> (ch <-> th))
9	8	rexbidv 2636	. . 3 |- (y = B -> (E.z e. T ch <-> E.z e. T th))
10	7, 9	rspc2ev 2964	. 2 |- ((A e. R /\ B e. S /\ E.z e. T th) -> E.x e. R E.y e. S E.z e. T ph)
11	1, 2, 5, 10	syl3anc 1182	1 |- (((A e. R /\ B e. S /\ C e. T) /\ ps) -> E.x e. R E.y e. S E.z e. T ph)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  E.wrex 2616

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-3an 936  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-rex 2621  df-v 2862

This theorem is referenced by: (None)