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Theorem ax13fromc9 31909
Description: Derive ax-13 2026 from ax-c9 31895 and other older axioms.

This proof uses newer axioms ax-4 1652 and ax-6 1771, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 31889 and ax-c10 31891. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax13fromc9  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )

Proof of Theorem ax13fromc9
StepHypRef Expression
1 ax-c5 31888 . . . . . 6  |-  ( A. x  x  =  y  ->  x  =  y )
21con3i 135 . . . . 5  |-  ( -.  x  =  y  ->  -.  A. x  x  =  y )
32adantr 463 . . . 4  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  A. x  x  =  y )
4 equtrr 1821 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  z  ->  x  =  y )
)
54equcoms 1819 . . . . . . 7  |-  ( y  =  z  ->  (
x  =  z  ->  x  =  y )
)
65con3rr3 136 . . . . . 6  |-  ( -.  x  =  y  -> 
( y  =  z  ->  -.  x  =  z ) )
76imp 427 . . . . 5  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  x  =  z )
8 ax-c5 31888 . . . . 5  |-  ( A. x  x  =  z  ->  x  =  z )
97, 8nsyl 121 . . . 4  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  -.  A. x  x  =  z )
10 ax-c9 31895 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
113, 9, 10sylc 59 . . 3  |-  ( ( -.  x  =  y  /\  y  =  z )  ->  ( y  =  z  ->  A. x  y  =  z )
)
1211ex 432 . 2  |-  ( -.  x  =  y  -> 
( y  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
1312pm2.43d 47 1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-c5 31888  ax-c9 31895
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634
This theorem is referenced by: (None)
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