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Theorem ssopab2 4290
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 1756 . . . 4  |-  F/ x A. x A. y (
ph  ->  ps )
2 nfa1 1756 . . . . . 6  |-  F/ y A. y ( ph  ->  ps )
3 sp 1716 . . . . . . 7  |-  ( A. y ( ph  ->  ps )  ->  ( ph  ->  ps ) )
43anim2d 548 . . . . . 6  |-  ( A. y ( ph  ->  ps )  ->  ( (
z  =  <. x ,  y >.  /\  ph )  ->  ( z  = 
<. x ,  y >.  /\  ps ) ) )
52, 4eximd 1750 . . . . 5  |-  ( A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
65sps 1739 . . . 4  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
71, 6eximd 1750 . . 3  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
87ss2abdv 3246 . 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 4078 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
10 df-opab 4078 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
118, 9, 103sstr4g 3219 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 1527   E.wex 1528    = wceq 1623   {cab 2269    C_ wss 3152   <.cop 3643   {copab 4076
This theorem is referenced by:  ssopab2b  4291  ssopab2i  4292  ssopab2dv  4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-in 3159  df-ss 3166  df-opab 4078
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