Theorem rspc3v 2964

Description: 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)

Hypotheses

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

Assertion

Ref	Expression
rspc3v	|- ((A e. R /\ B e. S /\ C e. T) -> (A.x e. R A.y e. S A.z e. T ph -> ps))

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 rspc3v

Step	Hyp	Ref	Expression
1		rspc3v.1	. . . . 5 |- (x = A -> (ph <-> ch))
2	1	ralbidv 2634	. . . 4 |- (x = A -> (A.z e. T ph <-> A.z e. T ch))
3		rspc3v.2	. . . . 5 |- (y = B -> (ch <-> th))
4	3	ralbidv 2634	. . . 4 |- (y = B -> (A.z e. T ch <-> A.z e. T th))
5	2, 4	rspc2v 2961	. . 3 |- ((A e. R /\ B e. S) -> (A.x e. R A.y e. S A.z e. T ph -> A.z e. T th))
6		rspc3v.3	. . . 4 |- (z = C -> (th <-> ps))
7	6	rspcv 2951	. . 3 |- (C e. T -> (A.z e. T th -> ps))
8	5, 7	sylan9 638	. 2 |- (((A e. R /\ B e. S) /\ C e. T) -> (A.x e. R A.y e. S A.z e. T ph -> ps))
9	8	3impa 1146	1 |- ((A e. R /\ B e. S /\ C e. T) -> (A.x e. R A.y e. S A.z e. T ph -> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  A.wral 2614

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 2478  df-ral 2619  df-v 2861

This theorem is referenced by:  caovassg  5626  caovdig  5632  caovdirg  5633  trd  5921