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Theorem wl-ax11-lem10 31835
Description: We now have prepared everything. The unwanted variable 
u is just in one place left. pm2.61 174 can be used in conjunction with wl-ax11-lem9 31834 to eliminate the second antecedent. Missing is something along the lines of ax-6 1794, 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 31833 . . . . 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 31831 . . . . 5  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  ( A. u A. x [ u  /  y ] ph  <->  A. x A. y ph ) )
31, 2bitr3d 258 . . . 4  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  ( A. y A. x ph  <->  A. x A. y ph ) )
43biimpd 210 . . 3  |-  ( ( A. u  u  =  y  /\  -.  A. x  x  =  y
)  ->  ( A. y A. x ph  ->  A. x A. y ph ) )
54ex 435 . 2  |-  ( A. u  u  =  y  ->  ( -.  A. x  x  =  y  ->  ( A. y A. x ph  ->  A. x A. y ph ) ) )
65aecoms 2107 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 370   A.wal 1435   [wsb 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905  ax-13 2053  ax-wl-11v 31825
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787
This theorem is referenced by: (None)
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