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Theorem prel12g 4212
Description: Closed form of prel12 4209. (Contributed by AV, 9-Dec-2018.)
Assertion
Ref Expression
prel12g  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( -.  A  =  B  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) ) )

Proof of Theorem prel12g
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2461 . . . . . . 7  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
21notbid 294 . . . . . 6  |-  ( x  =  A  ->  ( -.  x  =  y  <->  -.  A  =  y ) )
3 preq1 4111 . . . . . . . 8  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
43eqeq1d 2459 . . . . . . 7  |-  ( x  =  A  ->  ( { x ,  y }  =  { z ,  D }  <->  { A ,  y }  =  { z ,  D } ) )
5 eleq1 2529 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  { z ,  D }  <->  A  e.  { z ,  D }
) )
65anbi1d 704 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  {
z ,  D }  /\  y  e.  { z ,  D } )  <-> 
( A  e.  {
z ,  D }  /\  y  e.  { z ,  D } ) ) )
74, 6bibi12d 321 . . . . . 6  |-  ( x  =  A  ->  (
( { x ,  y }  =  {
z ,  D }  <->  ( x  e.  { z ,  D }  /\  y  e.  { z ,  D } ) )  <-> 
( { A , 
y }  =  {
z ,  D }  <->  ( A  e.  { z ,  D }  /\  y  e.  { z ,  D } ) ) ) )
82, 7imbi12d 320 . . . . 5  |-  ( x  =  A  ->  (
( -.  x  =  y  ->  ( {
x ,  y }  =  { z ,  D }  <->  ( x  e.  { z ,  D }  /\  y  e.  {
z ,  D }
) ) )  <->  ( -.  A  =  y  ->  ( { A ,  y }  =  { z ,  D }  <->  ( A  e.  { z ,  D }  /\  y  e.  {
z ,  D }
) ) ) ) )
98imbi2d 316 . . . 4  |-  ( x  =  A  ->  (
( D  e.  Y  ->  ( -.  x  =  y  ->  ( {
x ,  y }  =  { z ,  D }  <->  ( x  e.  { z ,  D }  /\  y  e.  {
z ,  D }
) ) ) )  <-> 
( D  e.  Y  ->  ( -.  A  =  y  ->  ( { A ,  y }  =  { z ,  D } 
<->  ( A  e.  {
z ,  D }  /\  y  e.  { z ,  D } ) ) ) ) ) )
10 eqeq2 2472 . . . . . . 7  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
1110notbid 294 . . . . . 6  |-  ( y  =  B  ->  ( -.  A  =  y  <->  -.  A  =  B ) )
12 preq2 4112 . . . . . . . 8  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
1312eqeq1d 2459 . . . . . . 7  |-  ( y  =  B  ->  ( { A ,  y }  =  { z ,  D }  <->  { A ,  B }  =  {
z ,  D }
) )
14 eleq1 2529 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  { z ,  D }  <->  B  e.  { z ,  D }
) )
1514anbi2d 703 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  {
z ,  D }  /\  y  e.  { z ,  D } )  <-> 
( A  e.  {
z ,  D }  /\  B  e.  { z ,  D } ) ) )
1613, 15bibi12d 321 . . . . . 6  |-  ( y  =  B  ->  (
( { A , 
y }  =  {
z ,  D }  <->  ( A  e.  { z ,  D }  /\  y  e.  { z ,  D } ) )  <-> 
( { A ,  B }  =  {
z ,  D }  <->  ( A  e.  { z ,  D }  /\  B  e.  { z ,  D } ) ) ) )
1711, 16imbi12d 320 . . . . 5  |-  ( y  =  B  ->  (
( -.  A  =  y  ->  ( { A ,  y }  =  { z ,  D } 
<->  ( A  e.  {
z ,  D }  /\  y  e.  { z ,  D } ) ) )  <->  ( -.  A  =  B  ->  ( { A ,  B }  =  { z ,  D }  <->  ( A  e.  { z ,  D }  /\  B  e.  {
z ,  D }
) ) ) ) )
1817imbi2d 316 . . . 4  |-  ( y  =  B  ->  (
( D  e.  Y  ->  ( -.  A  =  y  ->  ( { A ,  y }  =  { z ,  D } 
<->  ( A  e.  {
z ,  D }  /\  y  e.  { z ,  D } ) ) ) )  <->  ( D  e.  Y  ->  ( -.  A  =  B  -> 
( { A ,  B }  =  {
z ,  D }  <->  ( A  e.  { z ,  D }  /\  B  e.  { z ,  D } ) ) ) ) ) )
19 preq1 4111 . . . . . . . 8  |-  ( z  =  C  ->  { z ,  D }  =  { C ,  D }
)
2019eqeq2d 2471 . . . . . . 7  |-  ( z  =  C  ->  ( { A ,  B }  =  { z ,  D } 
<->  { A ,  B }  =  { C ,  D } ) )
2119eleq2d 2527 . . . . . . . 8  |-  ( z  =  C  ->  ( A  e.  { z ,  D }  <->  A  e.  { C ,  D }
) )
2219eleq2d 2527 . . . . . . . 8  |-  ( z  =  C  ->  ( B  e.  { z ,  D }  <->  B  e.  { C ,  D }
) )
2321, 22anbi12d 710 . . . . . . 7  |-  ( z  =  C  ->  (
( A  e.  {
z ,  D }  /\  B  e.  { z ,  D } )  <-> 
( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) )
2420, 23bibi12d 321 . . . . . 6  |-  ( z  =  C  ->  (
( { A ,  B }  =  {
z ,  D }  <->  ( A  e.  { z ,  D }  /\  B  e.  { z ,  D } ) )  <-> 
( { A ,  B }  =  { C ,  D }  <->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) ) )
2524imbi2d 316 . . . . 5  |-  ( z  =  C  ->  (
( -.  A  =  B  ->  ( { A ,  B }  =  { z ,  D } 
<->  ( A  e.  {
z ,  D }  /\  B  e.  { z ,  D } ) ) )  <->  ( -.  A  =  B  ->  ( { A ,  B }  =  { C ,  D }  <->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D }
) ) ) ) )
2625imbi2d 316 . . . 4  |-  ( z  =  C  ->  (
( D  e.  Y  ->  ( -.  A  =  B  ->  ( { A ,  B }  =  { z ,  D } 
<->  ( A  e.  {
z ,  D }  /\  B  e.  { z ,  D } ) ) ) )  <->  ( D  e.  Y  ->  ( -.  A  =  B  -> 
( { A ,  B }  =  { C ,  D }  <->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) ) ) ) )
27 preq2 4112 . . . . . . . . 9  |-  ( w  =  D  ->  { z ,  w }  =  { z ,  D } )
2827eqeq2d 2471 . . . . . . . 8  |-  ( w  =  D  ->  ( { x ,  y }  =  { z ,  w }  <->  { x ,  y }  =  { z ,  D } ) )
2927eleq2d 2527 . . . . . . . . 9  |-  ( w  =  D  ->  (
x  e.  { z ,  w }  <->  x  e.  { z ,  D }
) )
3027eleq2d 2527 . . . . . . . . 9  |-  ( w  =  D  ->  (
y  e.  { z ,  w }  <->  y  e.  { z ,  D }
) )
3129, 30anbi12d 710 . . . . . . . 8  |-  ( w  =  D  ->  (
( x  e.  {
z ,  w }  /\  y  e.  { z ,  w } )  <-> 
( x  e.  {
z ,  D }  /\  y  e.  { z ,  D } ) ) )
3228, 31bibi12d 321 . . . . . . 7  |-  ( w  =  D  ->  (
( { x ,  y }  =  {
z ,  w }  <->  ( x  e.  { z ,  w }  /\  y  e.  { z ,  w } ) )  <-> 
( { x ,  y }  =  {
z ,  D }  <->  ( x  e.  { z ,  D }  /\  y  e.  { z ,  D } ) ) ) )
3332imbi2d 316 . . . . . 6  |-  ( w  =  D  ->  (
( -.  x  =  y  ->  ( {
x ,  y }  =  { z ,  w }  <->  ( x  e.  { z ,  w }  /\  y  e.  {
z ,  w }
) ) )  <->  ( -.  x  =  y  ->  ( { x ,  y }  =  { z ,  D }  <->  ( x  e.  { z ,  D }  /\  y  e.  {
z ,  D }
) ) ) ) )
34 vex 3112 . . . . . . 7  |-  x  e. 
_V
35 vex 3112 . . . . . . 7  |-  y  e. 
_V
36 vex 3112 . . . . . . 7  |-  z  e. 
_V
37 vex 3112 . . . . . . 7  |-  w  e. 
_V
3834, 35, 36, 37prel12 4209 . . . . . 6  |-  ( -.  x  =  y  -> 
( { x ,  y }  =  {
z ,  w }  <->  ( x  e.  { z ,  w }  /\  y  e.  { z ,  w } ) ) )
3933, 38vtoclg 3167 . . . . 5  |-  ( D  e.  Y  ->  ( -.  x  =  y  ->  ( { x ,  y }  =  {
z ,  D }  <->  ( x  e.  { z ,  D }  /\  y  e.  { z ,  D } ) ) ) )
4039a1i 11 . . . 4  |-  ( ( x  e.  V  /\  y  e.  W  /\  z  e.  X )  ->  ( D  e.  Y  ->  ( -.  x  =  y  ->  ( {
x ,  y }  =  { z ,  D }  <->  ( x  e.  { z ,  D }  /\  y  e.  {
z ,  D }
) ) ) ) )
419, 18, 26, 40vtocl3ga 3177 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( D  e.  Y  ->  ( -.  A  =  B  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) ) ) )
42413expa 1196 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( D  e.  Y  ->  ( -.  A  =  B  ->  ( { A ,  B }  =  { C ,  D }  <->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) ) ) )
4342impr 619 1  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( -.  A  =  B  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cpr 4034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-sn 4033  df-pr 4035
This theorem is referenced by:  hash2prd  12522
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