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Theorem ssopab2 4472
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 1806 . . . 4  |-  F/ x A. x A. y (
ph  ->  ps )
2 nfa1 1806 . . . . . 6  |-  F/ y A. y ( ph  ->  ps )
3 sp 1763 . . . . . . 7  |-  ( A. y ( ph  ->  ps )  ->  ( ph  ->  ps ) )
43anim2d 549 . . . . . 6  |-  ( A. y ( ph  ->  ps )  ->  ( (
z  =  <. x ,  y >.  /\  ph )  ->  ( z  = 
<. x ,  y >.  /\  ps ) ) )
52, 4eximd 1786 . . . . 5  |-  ( A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
65sps 1770 . . . 4  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
71, 6eximd 1786 . . 3  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
87ss2abdv 3408 . 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 4259 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
10 df-opab 4259 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
118, 9, 103sstr4g 3381 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 359   A.wal 1549   E.wex 1550    = wceq 1652   {cab 2421    C_ wss 3312   <.cop 3809   {copab 4257
This theorem is referenced by:  ssopab2b  4473  ssopab2i  4474  ssopab2dv  4475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-in 3319  df-ss 3326  df-opab 4259
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