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Theorem ssopab2 4414
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 1796 . . . 4  |-  F/ x A. x A. y (
ph  ->  ps )
2 nfa1 1796 . . . . . 6  |-  F/ y A. y ( ph  ->  ps )
3 sp 1755 . . . . . . 7  |-  ( A. y ( ph  ->  ps )  ->  ( ph  ->  ps ) )
43anim2d 549 . . . . . 6  |-  ( A. y ( ph  ->  ps )  ->  ( (
z  =  <. x ,  y >.  /\  ph )  ->  ( z  = 
<. x ,  y >.  /\  ps ) ) )
52, 4eximd 1778 . . . . 5  |-  ( A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
65sps 1762 . . . 4  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  ->  E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
71, 6eximd 1778 . . 3  |-  ( A. x A. y ( ph  ->  ps )  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
87ss2abdv 3352 . 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 4201 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
10 df-opab 4201 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
118, 9, 103sstr4g 3325 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 1546   E.wex 1547    = wceq 1649   {cab 2366    C_ wss 3256   <.cop 3753   {copab 4199
This theorem is referenced by:  ssopab2b  4415  ssopab2i  4416  ssopab2dv  4417
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-in 3263  df-ss 3270  df-opab 4201
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