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Theorem rspc3ev 3190
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
StepHypRef Expression
1 simpl1 991 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  A  e.  R )
2 simpl2 992 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  B  e.  S )
3 rspc3v.3 . . . 4  |-  ( z  =  C  ->  ( th 
<->  ps ) )
43rspcev 3179 . . 3  |-  ( ( C  e.  T  /\  ps )  ->  E. z  e.  T  th )
543ad2antl3 1152 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  E. z  e.  T  th )
6 rspc3v.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
76rexbidv 2868 . . 3  |-  ( x  =  A  ->  ( E. z  e.  T  ph  <->  E. z  e.  T  ch ) )
8 rspc3v.2 . . . 4  |-  ( y  =  B  ->  ( ch 
<->  th ) )
98rexbidv 2868 . . 3  |-  ( y  =  B  ->  ( E. z  e.  T  ch 
<->  E. z  e.  T  th ) )
107, 9rspc2ev 3188 . 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 )
111, 2, 5, 10syl3anc 1219 1  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  E. x  e.  R  E. y  e.  S  E. z  e.  T  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-v 3080
This theorem is referenced by:  f1dom3el3dif  6093  pmltpclem1  21067  axlowdim  23379  axeuclidlem  23380  br8d  26113  br8  27730  br6  27731  jm2.27  29525  3dim1lem5  33468  lplni2  33539
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