Theorem rspc3ev 2851

Description: 3-variable restricted existentional 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 995	. 2 |- ( ( ( A e. R /\ B e. S /\ C e. T ) /\ ps ) -> A e. R )
2		simpl2 996	. 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 2834	. . 3 |- ( ( C e. T /\ ps ) -> E. z e. T th )
5	4	3ad2antl3 1156	. 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 2471	. . 3 |- ( x = A -> ( E. z e. T ph <-> E. z e. T ch ) )
8		rspc3v.2	. . . 4 |- ( y = B -> ( ch <-> th ) )
9	8	rexbidv 2471	. . 3 |- ( y = B -> ( E. z e. T ch <-> E. z e. T th ) )
10	7, 9	rspc2ev 2849	. 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 1233	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   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732

This theorem is referenced by: (None)