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Theorem prel12g 4064
Description: Closed form of prel12 4061. (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 2449 . . . . . . 7  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
21notbid 294 . . . . . 6  |-  ( x  =  A  ->  ( -.  x  =  y  <->  -.  A  =  y ) )
3 preq1 3966 . . . . . . . 8  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
43eqeq1d 2451 . . . . . . 7  |-  ( x  =  A  ->  ( { x ,  y }  =  { z ,  D }  <->  { A ,  y }  =  { z ,  D } ) )
5 eleq1 2503 . . . . . . . 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 2452 . . . . . . 7  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
1110notbid 294 . . . . . 6  |-  ( y  =  B  ->  ( -.  A  =  y  <->  -.  A  =  B ) )
12 preq2 3967 . . . . . . . 8  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
1312eqeq1d 2451 . . . . . . 7  |-  ( y  =  B  ->  ( { A ,  y }  =  { z ,  D }  <->  { A ,  B }  =  {
z ,  D }
) )
14 eleq1 2503 . . . . . . . 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 3966 . . . . . . . 8  |-  ( z  =  C  ->  { z ,  D }  =  { C ,  D }
)
2019eqeq2d 2454 . . . . . . 7  |-  ( z  =  C  ->  ( { A ,  B }  =  { z ,  D } 
<->  { A ,  B }  =  { C ,  D } ) )
2119eleq2d 2510 . . . . . . . 8  |-  ( z  =  C  ->  ( A  e.  { z ,  D }  <->  A  e.  { C ,  D }
) )
2219eleq2d 2510 . . . . . . . 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 id 22 . . . . . 6  |-  ( D  e.  Y  ->  D  e.  Y )
28 preq2 3967 . . . . . . . . . 10  |-  ( w  =  D  ->  { z ,  w }  =  { z ,  D } )
2928eqeq2d 2454 . . . . . . . . 9  |-  ( w  =  D  ->  ( { x ,  y }  =  { z ,  w }  <->  { x ,  y }  =  { z ,  D } ) )
3028eleq2d 2510 . . . . . . . . . 10  |-  ( w  =  D  ->  (
x  e.  { z ,  w }  <->  x  e.  { z ,  D }
) )
3128eleq2d 2510 . . . . . . . . . 10  |-  ( w  =  D  ->  (
y  e.  { z ,  w }  <->  y  e.  { z ,  D }
) )
3230, 31anbi12d 710 . . . . . . . . 9  |-  ( w  =  D  ->  (
( x  e.  {
z ,  w }  /\  y  e.  { z ,  w } )  <-> 
( x  e.  {
z ,  D }  /\  y  e.  { z ,  D } ) ) )
3329, 32bibi12d 321 . . . . . . . 8  |-  ( 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 } ) ) ) )
3433imbi2d 316 . . . . . . 7  |-  ( 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 }
) ) ) ) )
3534adantl 466 . . . . . 6  |-  ( ( D  e.  Y  /\  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 }
) ) ) ) )
36 vex 2987 . . . . . . . 8  |-  x  e. 
_V
37 vex 2987 . . . . . . . 8  |-  y  e. 
_V
38 vex 2987 . . . . . . . 8  |-  z  e. 
_V
39 vex 2987 . . . . . . . 8  |-  w  e. 
_V
4036, 37, 38, 39prel12 4061 . . . . . . 7  |-  ( -.  x  =  y  -> 
( { x ,  y }  =  {
z ,  w }  <->  ( x  e.  { z ,  w }  /\  y  e.  { z ,  w } ) ) )
4140a1i 11 . . . . . 6  |-  ( D  e.  Y  ->  ( -.  x  =  y  ->  ( { x ,  y }  =  {
z ,  w }  <->  ( x  e.  { z ,  w }  /\  y  e.  { z ,  w } ) ) ) )
4227, 35, 41vtocld 3034 . . . . 5  |-  ( D  e.  Y  ->  ( -.  x  =  y  ->  ( { x ,  y }  =  {
z ,  D }  <->  ( x  e.  { z ,  D }  /\  y  e.  { z ,  D } ) ) ) )
4342a1i 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 }
) ) ) ) )
449, 18, 26, 43vtocl3ga 3052 . . 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 } ) ) ) ) )
45443expa 1187 . 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 } ) ) ) ) )
4645impr 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 965    = wceq 1369    e. wcel 1756   {cpr 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2986  df-un 3345  df-sn 3890  df-pr 3892
This theorem is referenced by:  hash2prd  12193
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