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Theorem rspc3ev 2778
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 967 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  A  e.  R )
2 simpl2 968 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  B  e.  S )
3 rspc3v.3 . . . 4  |-  ( z  =  C  ->  ( th 
<->  ps ) )
43rspcev 2761 . . 3  |-  ( ( C  e.  T  /\  ps )  ->  E. z  e.  T  th )
543ad2antl3 1128 . 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 2413 . . 3  |-  ( x  =  A  ->  ( E. z  e.  T  ph  <->  E. z  e.  T  ch ) )
8 rspc3v.2 . . . 4  |-  ( y  =  B  ->  ( ch 
<->  th ) )
98rexbidv 2413 . . 3  |-  ( y  =  B  ->  ( E. z  e.  T  ch 
<->  E. z  e.  T  th ) )
107, 9rspc2ev 2776 . 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 1199 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    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660
This theorem is referenced by: (None)
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