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Theorem preq12bg 4195
Description: Closed form of preq12b 4192. (Contributed by Scott Fenton, 28-Mar-2014.)
Assertion
Ref Expression
preq12bg  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( { A ,  B }  =  { C ,  D }  <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) )

Proof of Theorem preq12bg
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 4095 . . . . . . 7  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21eqeq1d 2456 . . . . . 6  |-  ( x  =  A  ->  ( { x ,  y }  =  { z ,  D }  <->  { A ,  y }  =  { z ,  D } ) )
3 eqeq1 2458 . . . . . . . 8  |-  ( x  =  A  ->  (
x  =  z  <->  A  =  z ) )
43anbi1d 702 . . . . . . 7  |-  ( x  =  A  ->  (
( x  =  z  /\  y  =  D )  <->  ( A  =  z  /\  y  =  D ) ) )
5 eqeq1 2458 . . . . . . . 8  |-  ( x  =  A  ->  (
x  =  D  <->  A  =  D ) )
65anbi1d 702 . . . . . . 7  |-  ( x  =  A  ->  (
( x  =  D  /\  y  =  z )  <->  ( A  =  D  /\  y  =  z ) ) )
74, 6orbi12d 707 . . . . . 6  |-  ( x  =  A  ->  (
( ( x  =  z  /\  y  =  D )  \/  (
x  =  D  /\  y  =  z )
)  <->  ( ( A  =  z  /\  y  =  D )  \/  ( A  =  D  /\  y  =  z )
) ) )
82, 7bibi12d 319 . . . . 5  |-  ( x  =  A  ->  (
( { x ,  y }  =  {
z ,  D }  <->  ( ( x  =  z  /\  y  =  D )  \/  ( x  =  D  /\  y  =  z ) ) )  <->  ( { A ,  y }  =  { z ,  D } 
<->  ( ( A  =  z  /\  y  =  D )  \/  ( A  =  D  /\  y  =  z )
) ) ) )
98imbi2d 314 . . . 4  |-  ( x  =  A  ->  (
( D  e.  Y  ->  ( { x ,  y }  =  {
z ,  D }  <->  ( ( x  =  z  /\  y  =  D )  \/  ( x  =  D  /\  y  =  z ) ) ) )  <->  ( D  e.  Y  ->  ( { A ,  y }  =  { z ,  D }  <->  ( ( A  =  z  /\  y  =  D )  \/  ( A  =  D  /\  y  =  z ) ) ) ) ) )
10 preq2 4096 . . . . . . 7  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
1110eqeq1d 2456 . . . . . 6  |-  ( y  =  B  ->  ( { A ,  y }  =  { z ,  D }  <->  { A ,  B }  =  {
z ,  D }
) )
12 eqeq1 2458 . . . . . . . 8  |-  ( y  =  B  ->  (
y  =  D  <->  B  =  D ) )
1312anbi2d 701 . . . . . . 7  |-  ( y  =  B  ->  (
( A  =  z  /\  y  =  D )  <->  ( A  =  z  /\  B  =  D ) ) )
14 eqeq1 2458 . . . . . . . 8  |-  ( y  =  B  ->  (
y  =  z  <->  B  =  z ) )
1514anbi2d 701 . . . . . . 7  |-  ( y  =  B  ->  (
( A  =  D  /\  y  =  z )  <->  ( A  =  D  /\  B  =  z ) ) )
1613, 15orbi12d 707 . . . . . 6  |-  ( y  =  B  ->  (
( ( A  =  z  /\  y  =  D )  \/  ( A  =  D  /\  y  =  z )
)  <->  ( ( A  =  z  /\  B  =  D )  \/  ( A  =  D  /\  B  =  z )
) ) )
1711, 16bibi12d 319 . . . . 5  |-  ( y  =  B  ->  (
( { A , 
y }  =  {
z ,  D }  <->  ( ( A  =  z  /\  y  =  D )  \/  ( A  =  D  /\  y  =  z ) ) )  <->  ( { A ,  B }  =  {
z ,  D }  <->  ( ( A  =  z  /\  B  =  D )  \/  ( A  =  D  /\  B  =  z ) ) ) ) )
1817imbi2d 314 . . . 4  |-  ( y  =  B  ->  (
( D  e.  Y  ->  ( { A , 
y }  =  {
z ,  D }  <->  ( ( A  =  z  /\  y  =  D )  \/  ( A  =  D  /\  y  =  z ) ) ) )  <->  ( D  e.  Y  ->  ( { A ,  B }  =  { z ,  D } 
<->  ( ( A  =  z  /\  B  =  D )  \/  ( A  =  D  /\  B  =  z )
) ) ) ) )
19 preq1 4095 . . . . . . 7  |-  ( z  =  C  ->  { z ,  D }  =  { C ,  D }
)
2019eqeq2d 2468 . . . . . 6  |-  ( z  =  C  ->  ( { A ,  B }  =  { z ,  D } 
<->  { A ,  B }  =  { C ,  D } ) )
21 eqeq2 2469 . . . . . . . 8  |-  ( z  =  C  ->  ( A  =  z  <->  A  =  C ) )
2221anbi1d 702 . . . . . . 7  |-  ( z  =  C  ->  (
( A  =  z  /\  B  =  D )  <->  ( A  =  C  /\  B  =  D ) ) )
23 eqeq2 2469 . . . . . . . 8  |-  ( z  =  C  ->  ( B  =  z  <->  B  =  C ) )
2423anbi2d 701 . . . . . . 7  |-  ( z  =  C  ->  (
( A  =  D  /\  B  =  z )  <->  ( A  =  D  /\  B  =  C ) ) )
2522, 24orbi12d 707 . . . . . 6  |-  ( z  =  C  ->  (
( ( A  =  z  /\  B  =  D )  \/  ( A  =  D  /\  B  =  z )
)  <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) ) )
2620, 25bibi12d 319 . . . . 5  |-  ( z  =  C  ->  (
( { A ,  B }  =  {
z ,  D }  <->  ( ( A  =  z  /\  B  =  D )  \/  ( A  =  D  /\  B  =  z ) ) )  <->  ( { A ,  B }  =  { C ,  D }  <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) ) )
2726imbi2d 314 . . . 4  |-  ( z  =  C  ->  (
( D  e.  Y  ->  ( { A ,  B }  =  {
z ,  D }  <->  ( ( A  =  z  /\  B  =  D )  \/  ( A  =  D  /\  B  =  z ) ) ) )  <->  ( D  e.  Y  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) ) ) ) )
28 preq2 4096 . . . . . . 7  |-  ( w  =  D  ->  { z ,  w }  =  { z ,  D } )
2928eqeq2d 2468 . . . . . 6  |-  ( w  =  D  ->  ( { x ,  y }  =  { z ,  w }  <->  { x ,  y }  =  { z ,  D } ) )
30 eqeq2 2469 . . . . . . . 8  |-  ( w  =  D  ->  (
y  =  w  <->  y  =  D ) )
3130anbi2d 701 . . . . . . 7  |-  ( w  =  D  ->  (
( x  =  z  /\  y  =  w )  <->  ( x  =  z  /\  y  =  D ) ) )
32 eqeq2 2469 . . . . . . . 8  |-  ( w  =  D  ->  (
x  =  w  <->  x  =  D ) )
3332anbi1d 702 . . . . . . 7  |-  ( w  =  D  ->  (
( x  =  w  /\  y  =  z )  <->  ( x  =  D  /\  y  =  z ) ) )
3431, 33orbi12d 707 . . . . . 6  |-  ( w  =  D  ->  (
( ( x  =  z  /\  y  =  w )  \/  (
x  =  w  /\  y  =  z )
)  <->  ( ( x  =  z  /\  y  =  D )  \/  (
x  =  D  /\  y  =  z )
) ) )
35 vex 3109 . . . . . . 7  |-  x  e. 
_V
36 vex 3109 . . . . . . 7  |-  y  e. 
_V
37 vex 3109 . . . . . . 7  |-  z  e. 
_V
38 vex 3109 . . . . . . 7  |-  w  e. 
_V
3935, 36, 37, 38preq12b 4192 . . . . . 6  |-  ( { x ,  y }  =  { z ,  w }  <->  ( (
x  =  z  /\  y  =  w )  \/  ( x  =  w  /\  y  =  z ) ) )
4029, 34, 39vtoclbg 3165 . . . . 5  |-  ( D  e.  Y  ->  ( { x ,  y }  =  { z ,  D }  <->  ( (
x  =  z  /\  y  =  D )  \/  ( x  =  D  /\  y  =  z ) ) ) )
4140a1i 11 . . . 4  |-  ( ( x  e.  V  /\  y  e.  W  /\  z  e.  X )  ->  ( D  e.  Y  ->  ( { x ,  y }  =  {
z ,  D }  <->  ( ( x  =  z  /\  y  =  D )  \/  ( x  =  D  /\  y  =  z ) ) ) ) )
429, 18, 27, 41vtocl3ga 3174 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( D  e.  Y  ->  ( { A ,  B }  =  { C ,  D }  <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) ) )
43423expa 1194 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( D  e.  Y  ->  ( { A ,  B }  =  { C ,  D }  <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) ) )
4443impr 617 1  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( { A ,  B }  =  { C ,  D }  <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-sn 4017  df-pr 4019
This theorem is referenced by:  prneimg  4197  pythagtriplem2  14425  pythagtrip  14442  usgraidx2v  24595  usgra2adedgspthlem2  24814  usgra2wlkspthlem1  24821  constr3trllem2  24853  preqsnd  27624  pr1eqbg  32671  usgedgvadf1  32787  usgedgvadf1ALT  32790
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