Theorem rspc3v 2937

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 2542	. . . 4 |- ( x = A -> ( A. z e. T ph <-> A. z e. T ch ) )
3		rspc3v.2	. . . . 5 |- ( y = B -> ( ch <-> th ) )
4	3	ralbidv 2542	. . . 4 |- ( y = B -> ( A. z e. T ch <-> A. z e. T th ) )
5	2, 4	rspc2v 2934	. . 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 2917	. . 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 1221	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 1005   = wceq 1398   e. wcel 2203   A.wral 2520

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815

This theorem is referenced by:  swopolem  4426  isopolem  5995  isosolem  5997  caovassg  6213  caovcang  6216  caovordig  6220  caovordg  6222  caovdig  6229  caovdirg  6232  caoftrn  6299  sgrpass  13621  rngdi  14084  rngdir  14085  islmodd  14441  rmodislmodlem  14498  rmodislmod  14499  lssclg  14512  psmettri2  15193  xmettri2  15226