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Theorem ssoprab2i 6371
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 567 . . . 4  |-  ( ( w  =  <. x ,  y >.  /\  ph )  ->  ( w  = 
<. x ,  y >.  /\  ps ) )
322eximi 1678 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( w  =  <. x ,  y >.  /\  ps ) )
43ssopab2i 4717 . 2  |-  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) } 
C_  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ps ) }
5 dfoprab2 6323 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
6 dfoprab2 6323 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ps ) }
74, 5, 63sstr4i 3480 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  C_  { <. <. x ,  y >. ,  z
>.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405   E.wex 1633    C_ wss 3413   <.cop 3977   {copab 4451   {coprab 6278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-opab 4453  df-oprab 6281
This theorem is referenced by:  sxbrsigalem5  28722
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