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Theorem ssoprab2g 24430
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
Dummy variable  u is distinct from all other variables.
Allowed substitution hints:    ps( x, y, z)    ch( x, y, z)

Proof of Theorem ssoprab2g
StepHypRef Expression
1 ssoprab2g.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
21anim2d 550 . . . 4  |-  ( ph  ->  ( ( u  = 
<. x ,  y >.  /\  ps )  ->  (
u  =  <. x ,  y >.  /\  ch ) ) )
322eximdv 1611 . . 3  |-  ( ph  ->  ( E. x E. y ( u  = 
<. x ,  y >.  /\  ps )  ->  E. x E. y ( u  = 
<. x ,  y >.  /\  ch ) ) )
43ssopab2dv 4292 . 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 5856 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ps ) }
6 dfoprab2 5856 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ch }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ch ) }
74, 5, 63sstr4g 3220 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  C_  { <. <.
x ,  y >. ,  z >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1529    = wceq 1624    C_ wss 3153   <.cop 3644   {copab 4077   {coprab 5820
This theorem is referenced by:  oprabex2gpop  24434
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-opab 4079  df-oprab 5823
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