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Theorem ssoprab2i 6102
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 553 . . . 4  |-  ( ( w  =  <. x ,  y >.  /\  ph )  ->  ( w  = 
<. x ,  y >.  /\  ps ) )
322eximi 1583 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( w  =  <. x ,  y >.  /\  ps ) )
43ssopab2i 4424 . 2  |-  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) } 
C_  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ps ) }
5 dfoprab2 6061 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
6 dfoprab2 6061 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ps ) }
74, 5, 63sstr4i 3331 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  C_  { <. <. x ,  y >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    C_ wss 3264   <.cop 3761   {copab 4207   {coprab 6022
This theorem is referenced by:  sxbrsigalem5  24433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-opab 4209  df-oprab 6025
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