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Theorem ssoprab2i 5645
Description: Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
ssoprab2i.1  |-  ( ph  ->  ps )
Assertion
Ref Expression
ssoprab2i  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  C_  { <. <. x ,  y >. ,  z
>.  |  ps }
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem ssoprab2i
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ssoprab2i.1 . . . . 5  |-  ( ph  ->  ps )
21anim2i 334 . . . 4  |-  ( ( w  =  <. x ,  y >.  /\  ph )  ->  ( w  = 
<. x ,  y >.  /\  ps ) )
322eximi 1533 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( w  =  <. x ,  y >.  /\  ps ) )
43ssopab2i 4061 . 2  |-  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) } 
C_  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ps ) }
5 dfoprab2 5604 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
6 dfoprab2 5604 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ps ) }
74, 5, 63sstr4i 3048 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  C_  { <. <. x ,  y >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    C_ wss 2983   <.cop 3420   {copab 3859   {coprab 5565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-opab 3861  df-oprab 5568
This theorem is referenced by: (None)
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