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Theorem wl-aleq 31832
Description: The semantics of  A. x
y  =  z. (Contributed by Wolf Lammen, 27-Apr-2018.)
Assertion
Ref Expression
wl-aleq  |-  ( A. x  y  =  z  <->  ( y  =  z  /\  ( A. x  x  =  y  <->  A. x  x  =  z ) ) )

Proof of Theorem wl-aleq
StepHypRef Expression
1 sp 1914 . . 3  |-  ( A. x  y  =  z  ->  y  =  z )
2 equequ2 1853 . . . . 5  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
32alimi 1678 . . . 4  |-  ( A. x  y  =  z  ->  A. x ( x  =  y  <->  x  =  z ) )
4 albi 1684 . . . 4  |-  ( A. x ( x  =  y  <->  x  =  z
)  ->  ( A. x  x  =  y  <->  A. x  x  =  z ) )
53, 4syl 17 . . 3  |-  ( A. x  y  =  z  ->  ( A. x  x  =  y  <->  A. x  x  =  z )
)
61, 5jca 534 . 2  |-  ( A. x  y  =  z  ->  ( y  =  z  /\  ( A. x  x  =  y  <->  A. x  x  =  z )
) )
7 ax-7 1843 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
87al2imi 1681 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  ->  A. x  y  =  z )
)
98a1dd 47 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  ->  (
y  =  z  ->  A. x  y  =  z ) ) )
10 axc9 2105 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
119, 10bija 356 . . 3  |-  ( ( A. x  x  =  y  <->  A. x  x  =  z )  ->  (
y  =  z  ->  A. x  y  =  z ) )
1211impcom 431 . 2  |-  ( ( y  =  z  /\  ( A. x  x  =  y  <->  A. x  x  =  z ) )  ->  A. x  y  =  z )
136, 12impbii 190 1  |-  ( A. x  y  =  z  <->  ( y  =  z  /\  ( A. x  x  =  y  <->  A. x  x  =  z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435
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 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  wl-nfeqfb  31834  wl-ax11-lem2  31880
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