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Theorem wl-exeq 31774
Description: The semantics of  E. x
y  =  z. (Contributed by Wolf Lammen, 27-Apr-2018.)
Assertion
Ref Expression
wl-exeq  |-  ( E. x  y  =  z  <-> 
( y  =  z  \/  A. x  x  =  y  \/  A. x  x  =  z
) )

Proof of Theorem wl-exeq
StepHypRef Expression
1 nfeqf 2111 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x  y  =  z )
2119.9d 1945 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  ( E. x  y  =  z  ->  y  =  z ) )
32impancom 441 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  E. x  y  =  z
)  ->  ( -.  A. x  x  =  z  ->  y  =  z ) )
43orrd 379 . . . . 5  |-  ( ( -.  A. x  x  =  y  /\  E. x  y  =  z
)  ->  ( A. x  x  =  z  \/  y  =  z
) )
54expcom 436 . . . 4  |-  ( E. x  y  =  z  ->  ( -.  A. x  x  =  y  ->  ( A. x  x  =  z  \/  y  =  z ) ) )
65orrd 379 . . 3  |-  ( E. x  y  =  z  ->  ( A. x  x  =  y  \/  ( A. x  x  =  z  \/  y  =  z ) ) )
7 3orrot 988 . . . 4  |-  ( ( y  =  z  \/ 
A. x  x  =  y  \/  A. x  x  =  z )  <->  ( A. x  x  =  y  \/  A. x  x  =  z  \/  y  =  z )
)
8 3orass 985 . . . 4  |-  ( ( A. x  x  =  y  \/  A. x  x  =  z  \/  y  =  z )  <->  ( A. x  x  =  y  \/  ( A. x  x  =  z  \/  y  =  z
) ) )
97, 8bitri 252 . . 3  |-  ( ( y  =  z  \/ 
A. x  x  =  y  \/  A. x  x  =  z )  <->  ( A. x  x  =  y  \/  ( A. x  x  =  z  \/  y  =  z
) ) )
106, 9sylibr 215 . 2  |-  ( E. x  y  =  z  ->  ( y  =  z  \/  A. x  x  =  y  \/  A. x  x  =  z ) )
11 19.8a 1912 . . 3  |-  ( y  =  z  ->  E. x  y  =  z )
12 ax6e 2066 . . . . 5  |-  E. x  x  =  z
13 ax-7 1843 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
1413com12 32 . . . . 5  |-  ( x  =  z  ->  (
x  =  y  -> 
y  =  z ) )
1512, 14eximii 1703 . . . 4  |-  E. x
( x  =  y  ->  y  =  z )
161519.35i 1735 . . 3  |-  ( A. x  x  =  y  ->  E. x  y  =  z )
17 ax6e 2066 . . . . 5  |-  E. x  x  =  y
1817, 13eximii 1703 . . . 4  |-  E. x
( x  =  z  ->  y  =  z )
191819.35i 1735 . . 3  |-  ( A. x  x  =  z  ->  E. x  y  =  z )
2011, 16, 193jaoi 1327 . 2  |-  ( ( y  =  z  \/ 
A. x  x  =  y  \/  A. x  x  =  z )  ->  E. x  y  =  z )
2110, 20impbii 190 1  |-  ( E. x  y  =  z  <-> 
( y  =  z  \/  A. x  x  =  y  \/  A. x  x  =  z
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981   A.wal 1435   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2063
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-ex 1658  df-nf 1662
This theorem is referenced by:  wl-nfeqfb  31777
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