Theorem rspc3ev 3326

Description: 3-variable restricted existential 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 1064	. 2 |- ( ( ( A e. R /\ B e. S /\ C e. T ) /\ ps ) -> A e. R )
2		simpl2 1065	. 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 3309	. . 3 |- ( ( C e. T /\ ps ) -> E. z e. T th )
5	4	3ad2antl3 1225	. 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 3052	. . 3 |- ( x = A -> ( E. z e. T ph <-> E. z e. T ch ) )
8		rspc3v.2	. . . 4 |- ( y = B -> ( ch <-> th ) )
9	8	rexbidv 3052	. . 3 |- ( y = B -> ( E. z e. T ch <-> E. z e. T th ) )
10	7, 9	rspc2ev 3324	. 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 1326	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202

This theorem is referenced by:  f1dom3el3dif  6526  wrdl3s3  13705  pmltpclem1  23217  axlowdim  25841  axeuclidlem  25842  upgr3v3e3cycl  27040  br8d  29422  tgoldbachgt  30741  br8  31646  br6  31647  3dim1lem5  34752  lplni2  34823  jm2.27  37575