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Theorem rspc3ev 3170
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 998 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  A  e.  R )
2 simpl2 999 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)  /\  ps )  ->  B  e.  S )
3 rspc3v.3 . . . 4  |-  ( z  =  C  ->  ( th 
<->  ps ) )
43rspcev 3157 . . 3  |-  ( ( C  e.  T  /\  ps )  ->  E. z  e.  T  th )
543ad2antl3 1159 . 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 2915 . . 3  |-  ( x  =  A  ->  ( E. z  e.  T  ph  <->  E. z  e.  T  ch ) )
8 rspc3v.2 . . . 4  |-  ( y  =  B  ->  ( ch 
<->  th ) )
98rexbidv 2915 . . 3  |-  ( y  =  B  ->  ( E. z  e.  T  ch 
<->  E. z  e.  T  th ) )
107, 9rspc2ev 3168 . 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 1228 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840   E.wrex 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-rex 2757  df-v 3058
This theorem is referenced by:  f1dom3el3dif  6111  pmltpclem1  22042  axlowdim  24563  axeuclidlem  24564  br8d  27781  br8  29850  br6  29851  3dim1lem5  32447  lplni2  32518  jm2.27  35276
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