Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-aleq Structured version   Unicode version

Theorem wl-aleq 26270
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 1766 . . 3  |-  ( A. x  y  =  z  ->  y  =  z )
2 equequ2 1701 . . . . 5  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
32alimi 1569 . . . 4  |-  ( A. x  y  =  z  ->  A. x ( x  =  y  <->  x  =  z ) )
4 albi 1574 . . . 4  |-  ( A. x ( x  =  y  <->  x  =  z
)  ->  ( A. x  x  =  y  <->  A. x  x  =  z ) )
53, 4syl 16 . . 3  |-  ( A. x  y  =  z  ->  ( A. x  x  =  y  <->  A. x  x  =  z )
)
61, 5jca 520 . 2  |-  ( A. x  y  =  z  ->  ( y  =  z  /\  ( A. x  x  =  y  <->  A. x  x  =  z )
) )
7 ax-8 1690 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
87al2imi 1571 . . . . 5  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  ->  A. x  y  =  z )
)
98a1dd 45 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x  x  =  z  ->  (
y  =  z  ->  A. x  y  =  z ) ) )
10 ax12o 2014 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
119, 10bija 346 . . 3  |-  ( ( A. x  x  =  y  <->  A. x  x  =  z )  ->  (
y  =  z  ->  A. x  y  =  z ) )
1211impcom 421 . 2  |-  ( ( y  =  z  /\  ( A. x  x  =  y  <->  A. x  x  =  z ) )  ->  A. x  y  =  z )
136, 12impbii 182 1  |-  ( A. x  y  =  z  <->  ( y  =  z  /\  ( A. x  x  =  y  <->  A. x  x  =  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
  Copyright terms: Public domain W3C validator