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Theorem wl-ax11-lem10 28408
Description: We now have prepared everything. The unwanted variable 
u is just in one place left. pm2.61 171 can be used in conjunction with wl-ax11-lem9 28407 to eliminate the second antecedent. Missing is something along the lines of ax-6 1708, so we could remove the first antecedent. But the Metamath axioms cannot accomplish this. Such a rule must reside one abstraction level higher than all others: It says that a distinctor implies a distinct variable condition on its contained setvar. This is only needed if such conditions are required, as ax-11v does. The result of this study is for me, that you cannot introduce a setvar capturing this condition, and hope to eliminate it later. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem10  |-  ( A. y  y  =  u  ->  ( -.  A. x  x  =  y  ->  ( A. y A. x ph  ->  A. x A. y ph ) ) )
Distinct variable group:    x, u
Allowed substitution hints:    ph( x, y, u)

Proof of Theorem wl-ax11-lem10
StepHypRef Expression
1 wl-ax11-lem8 28406 . . . . 5  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  ( A. u A. x [ u  /  y ] ph  <->  A. y A. x ph ) )
2 wl-ax11-lem6 28404 . . . . 5  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  ( A. u A. x [ u  /  y ] ph  <->  A. x A. y ph ) )
31, 2bitr3d 255 . . . 4  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  ( A. y A. x ph  <->  A. x A. y ph ) )
43biimpd 207 . . 3  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  ( A. y A. x ph  ->  A. x A. y ph ) )
54ex 434 . 2  |-  ( A. u  u  =  y  ->  ( -.  A. x  x  =  y  ->  ( A. y A. x ph  ->  A. x A. y ph ) ) )
65aecoms 1999 1  |-  ( A. y  y  =  u  ->  ( -.  A. x  x  =  y  ->  ( A. y A. x ph  ->  A. x A. y ph ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1367   [wsb 1700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-12 1792  ax-13 1943  ax-wl-11v 28398
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701
This theorem is referenced by: (None)
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