Theorem rspc3v 2893

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 2506	. . . 4 |- ( x = A -> ( A. z e. T ph <-> A. z e. T ch ) )
3		rspc3v.2	. . . . 5 |- ( y = B -> ( ch <-> th ) )
4	3	ralbidv 2506	. . . 4 |- ( y = B -> ( A. z e. T ch <-> A. z e. T th ) )
5	2, 4	rspc2v 2890	. . 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 2873	. . 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 1197	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 981   = wceq 1373   e. wcel 2176   A.wral 2484

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774

This theorem is referenced by:  swopolem  4352  isopolem  5891  isosolem  5893  caovassg  6105  caovcang  6108  caovordig  6112  caovordg  6114  caovdig  6121  caovdirg  6124  caoftrn  6191  sgrpass  13240  rngdi  13702  rngdir  13703  islmodd  14055  rmodislmodlem  14112  rmodislmod  14113  lssclg  14126  psmettri2  14800  xmettri2  14833