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Theorem ssoprab2g 25032
Description: Inference of operation class abstraction subclass from implication. (Contributed by FL, 24-Jan-2010.)
Hypothesis
Ref Expression
ssoprab2g.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
ssoprab2g  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  C_  { <. <.
x ,  y >. ,  z >.  |  ch } )
Distinct variable groups:    ph, x, z    ph, y, z
Allowed substitution hints:    ps( x, y, z)    ch( x, y, z)

Proof of Theorem ssoprab2g
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 ssoprab2g.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
21anim2d 548 . . . 4  |-  ( ph  ->  ( ( u  = 
<. x ,  y >.  /\  ps )  ->  (
u  =  <. x ,  y >.  /\  ch ) ) )
322eximdv 1610 . . 3  |-  ( ph  ->  ( E. x E. y ( u  = 
<. x ,  y >.  /\  ps )  ->  E. x E. y ( u  = 
<. x ,  y >.  /\  ch ) ) )
43ssopab2dv 4293 . 2  |-  ( ph  ->  { <. u ,  z
>.  |  E. x E. y ( u  = 
<. x ,  y >.  /\  ps ) }  C_  {
<. u ,  z >.  |  E. x E. y
( u  =  <. x ,  y >.  /\  ch ) } )
5 dfoprab2 5895 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ps ) }
6 dfoprab2 5895 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ch }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ch ) }
74, 5, 63sstr4g 3219 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  C_  { <. <.
x ,  y >. ,  z >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    C_ wss 3152   <.cop 3643   {copab 4076   {coprab 5859
This theorem is referenced by:  oprabex2gpop  25036
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-oprab 5862
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