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Theorem ssoprab2i 6368
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 569 . . . 4  |-  ( ( w  =  <. x ,  y >.  /\  ph )  ->  ( w  = 
<. x ,  y >.  /\  ps ) )
322eximi 1631 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( w  =  <. x ,  y >.  /\  ps ) )
43ssopab2i 4770 . 2  |-  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) } 
C_  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ps ) }
5 dfoprab2 6320 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
6 dfoprab2 6320 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ps ) }
74, 5, 63sstr4i 3538 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  C_  { <. <. x ,  y >. ,  z
>.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374   E.wex 1591    C_ wss 3471   <.cop 4028   {copab 4499   {coprab 6278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-opab 4501  df-oprab 6281
This theorem is referenced by:  sxbrsigalem5  27887
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