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Theorem ssopab2 4779
Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
Assertion
Ref Expression
ssopab2  |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps } )

Proof of Theorem ssopab2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
21anim2d 565 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( ( z  = 
<. x ,  y >.  /\  ph )  ->  (
z  =  <. x ,  y >.  /\  ps ) ) )
32aleximi 1632 . . . 4  |-  ( A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
43aleximi 1632 . . 3  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
54ss2abdv 3578 . 2  |-  ( A. x A. y ( ph  ->  ps )  ->  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  C_  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) } )
6 df-opab 4512 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
7 df-opab 4512 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
85, 6, 73sstr4g 3550 1  |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596   {cab 2452    C_ wss 3481   <.cop 4039   {copab 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-in 3488  df-ss 3495  df-opab 4512
This theorem is referenced by:  ssopab2b  4780  ssopab2i  4781  ssopab2dv  4782
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