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Theorem ax10 1677
Description: Proof of axiom ax-10 1678 from others, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (See remarks for ax12o10lem1 1635 about why we use ax-9v 1632 instead of ax-9 1684.)

Our current practice is to use axiom ax-10 1678 from here on instead of theorem ax10 1677, in order to preferentially use ax-9 1684 instead of ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (New usage is discouraged.)

Assertion
Ref Expression
ax10  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Proof of Theorem ax10
StepHypRef Expression
1 ax-9v 1632 . 2  |-  -.  A. z  -.  z  =  x
2 df-ex 1538 . . 3  |-  ( E. z  z  =  x  <->  -.  A. z  -.  z  =  x )
3 ax-17 1628 . . . . 5  |-  ( ( A. x  x  =  y  ->  A. y 
y  =  x )  ->  A. z ( A. x  x  =  y  ->  A. y  y  =  x ) )
4 ax10lem25 1674 . . . . . . . . . 10  |-  ( -. 
A. y  y  =  x  ->  ( z  =  x  ->  A. y 
z  =  x ) )
54imp 420 . . . . . . . . 9  |-  ( ( -.  A. y  y  =  x  /\  z  =  x )  ->  A. y 
z  =  x )
6 ax10lem23 1672 . . . . . . . . . 10  |-  ( A. x  x  =  y  ->  ( A. y  z  =  x  ->  A. x  z  =  x )
)
7 ax12o10lem1 1635 . . . . . . . . . . 11  |-  ( z  =  x  ->  x  =  z )
87alimi 1546 . . . . . . . . . 10  |-  ( A. x  z  =  x  ->  A. x  x  =  z )
96, 8syl6 31 . . . . . . . . 9  |-  ( A. x  x  =  y  ->  ( A. y  z  =  x  ->  A. x  x  =  z )
)
10 ax10lem27 1676 . . . . . . . . 9  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
115, 9, 10syl56 32 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( ( -.  A. y  y  =  x  /\  z  =  x
)  ->  A. y 
y  =  x ) )
1211exp3acom23 1368 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( z  =  x  ->  ( -.  A. y  y  =  x  ->  A. y  y  =  x ) ) )
13 pm2.18 104 . . . . . . 7  |-  ( ( -.  A. y  y  =  x  ->  A. y 
y  =  x )  ->  A. y  y  =  x )
1412, 13syl6 31 . . . . . 6  |-  ( A. x  x  =  y  ->  ( z  =  x  ->  A. y  y  =  x ) )
1514com12 29 . . . . 5  |-  ( z  =  x  ->  ( A. x  x  =  y  ->  A. y  y  =  x ) )
163, 15ax10lem18 1667 . . . 4  |-  ( E. z  z  =  x  ->  ( A. x  x  =  y  ->  A. y  y  =  x ) )
1716com12 29 . . 3  |-  ( A. x  x  =  y  ->  ( E. z  z  =  x  ->  A. y 
y  =  x ) )
182, 17syl5bir 211 . 2  |-  ( A. x  x  =  y  ->  ( -.  A. z  -.  z  =  x  ->  A. y  y  =  x ) )
191, 18mpi 18 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   A.wal 1532   E.wex 1537
This theorem is referenced by:  e2ebindALT  27396
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-9v 1632  ax-12 1633
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538
  Copyright terms: Public domain W3C validator