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Theorem ssopab2 4306
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 nfa1 1768 . . . 4  |-  F/ x A. x A. y (
ph  ->  ps )
2 nfa1 1768 . . . . . 6  |-  F/ y A. y ( ph  ->  ps )
3 sp 1728 . . . . . . 7  |-  ( A. y ( ph  ->  ps )  ->  ( ph  ->  ps ) )
43anim2d 548 . . . . . 6  |-  ( A. y ( ph  ->  ps )  ->  ( (
z  =  <. x ,  y >.  /\  ph )  ->  ( z  = 
<. x ,  y >.  /\  ps ) ) )
52, 4eximd 1762 . . . . 5  |-  ( A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
65sps 1751 . . . 4  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
71, 6eximd 1762 . . 3  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
87ss2abdv 3259 . 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 ) } )
9 df-opab 4094 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
10 df-opab 4094 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
118, 9, 103sstr4g 3232 1  |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632   {cab 2282    C_ wss 3165   <.cop 3656   {copab 4092
This theorem is referenced by:  ssopab2b  4307  ssopab2i  4308  ssopab2dv  4309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-in 3172  df-ss 3179  df-opab 4094
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