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Theorem rspc3ev 2894
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
StepHypRef Expression
1 simpl1 958 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  A  e.  R )
2 simpl2 959 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  B  e.  S )
3 rspc3v.3 . . . 4  |-  ( z  =  C  ->  ( th 
<->  ps ) )
43rspcev 2884 . . 3  |-  ( ( C  e.  T  /\  ps )  ->  E. z  e.  T  th )
543ad2antl3 1119 . 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 2564 . . 3  |-  ( x  =  A  ->  ( E. z  e.  T  ph  <->  E. z  e.  T  ch ) )
8 rspc3v.2 . . . 4  |-  ( y  =  B  ->  ( ch 
<->  th ) )
98rexbidv 2564 . . 3  |-  ( y  =  B  ->  ( E. z  e.  T  ch 
<->  E. z  e.  T  th ) )
107, 9rspc2ev 2892 . 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 1182 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544
This theorem is referenced by:  pmltpclem1  18808  br8  24113  br6  24114  axlowdim  24589  axeuclidlem  24590  jm2.27  27101  3dim1lem5  29655  lplni2  29726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790
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