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Theorem ssoprab2 5920
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4306. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
ssoprab2  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  { <. <. x ,  y
>. ,  z >.  | 
ph }  C_  { <. <.
x ,  y >. ,  z >.  |  ps } )

Proof of Theorem ssoprab2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . . . . . 10  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
21anim2d 548 . . . . . . . . 9  |-  ( (
ph  ->  ps )  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
32alimi 1549 . . . . . . . 8  |-  ( A. z ( ph  ->  ps )  ->  A. z
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
4 exim 1565 . . . . . . . 8  |-  ( A. z ( ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) )  ->  ( E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph )  ->  E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ps ) ) )
53, 4syl 15 . . . . . . 7  |-  ( A. z ( ph  ->  ps )  ->  ( E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ps ) ) )
65alimi 1549 . . . . . 6  |-  ( A. y A. z ( ph  ->  ps )  ->  A. y
( E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) ) )
7 exim 1565 . . . . . 6  |-  ( A. y ( E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph )  ->  E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ps ) )  -> 
( E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) ) )
86, 7syl 15 . . . . 5  |-  ( A. y A. z ( ph  ->  ps )  ->  ( E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
98alimi 1549 . . . 4  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  A. x ( E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
10 exim 1565 . . . 4  |-  ( A. x ( E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) )  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
119, 10syl 15 . . 3  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  ( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
1211ss2abdv 3259 . 2  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  { w  |  E. x E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) }  C_  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) } )
13 df-oprab 5878 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
14 df-oprab 5878 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) }
1512, 13, 143sstr4g 3232 1  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  { <. <. x ,  y
>. ,  z >.  | 
ph }  C_  { <. <.
x ,  y >. ,  z >.  |  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   {coprab 5875
This theorem is referenced by:  ssoprab2b  5921
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-oprab 5878
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