Theorem rspc3v 2924

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 2530	. . . 4 |- ( x = A -> ( A. z e. T ph <-> A. z e. T ch ) )
3		rspc3v.2	. . . . 5 |- ( y = B -> ( ch <-> th ) )
4	3	ralbidv 2530	. . . 4 |- ( y = B -> ( A. z e. T ch <-> A. z e. T th ) )
5	2, 4	rspc2v 2921	. . 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 2904	. . 3 |- ( C e. T -> ( A. z e. T th -> ps ) )
8	5, 7	sylan9 409	. 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 1218	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   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2802

This theorem is referenced by:  swopolem  4400  isopolem  5958  isosolem  5960  caovassg  6176  caovcang  6179  caovordig  6183  caovordg  6185  caovdig  6192  caovdirg  6195  caoftrn  6263  sgrpass  13481  rngdi  13943  rngdir  13944  islmodd  14297  rmodislmodlem  14354  rmodislmod  14355  lssclg  14368  psmettri2  15042  xmettri2  15075