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Theorem aev-o 32210
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-c16 32172. Version of aev 2001 using ax-c11 32167. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
aev-o  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Distinct variable group:    x, y

Proof of Theorem aev-o
Dummy variables  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbae-o 32181 . 2  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
2 hbae-o 32181 . . . 4  |-  ( A. x  x  =  y  ->  A. t A. x  x  =  y )
3 ax-7 1841 . . . . 5  |-  ( x  =  t  ->  (
x  =  y  -> 
t  =  y ) )
43spimv 2065 . . . 4  |-  ( A. x  x  =  y  ->  t  =  y )
52, 4alrimih 1689 . . 3  |-  ( A. x  x  =  y  ->  A. t  t  =  y )
6 ax-7 1841 . . . . . . . 8  |-  ( y  =  u  ->  (
y  =  t  ->  u  =  t )
)
7 equcomi 1845 . . . . . . . 8  |-  ( u  =  t  ->  t  =  u )
86, 7syl6 34 . . . . . . 7  |-  ( y  =  u  ->  (
y  =  t  -> 
t  =  u ) )
98spimv 2065 . . . . . 6  |-  ( A. y  y  =  t  ->  t  =  u )
109aecoms-o 32180 . . . . 5  |-  ( A. t  t  =  y  ->  t  =  u )
1110axc4i-o 32178 . . . 4  |-  ( A. t  t  =  y  ->  A. t  t  =  u )
12 hbae-o 32181 . . . . 5  |-  ( A. t  t  =  u  ->  A. v A. t 
t  =  u )
13 ax-7 1841 . . . . . 6  |-  ( t  =  v  ->  (
t  =  u  -> 
v  =  u ) )
1413spimv 2065 . . . . 5  |-  ( A. t  t  =  u  ->  v  =  u )
1512, 14alrimih 1689 . . . 4  |-  ( A. t  t  =  u  ->  A. v  v  =  u )
16 aecom-o 32179 . . . 4  |-  ( A. v  v  =  u  ->  A. u  u  =  v )
1711, 15, 163syl 18 . . 3  |-  ( A. t  t  =  y  ->  A. u  u  =  v )
18 ax-7 1841 . . . 4  |-  ( u  =  w  ->  (
u  =  v  ->  w  =  v )
)
1918spimv 2065 . . 3  |-  ( A. u  u  =  v  ->  w  =  v )
205, 17, 193syl 18 . 2  |-  ( A. x  x  =  y  ->  w  =  v )
211, 20alrimih 1689 1  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435
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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-c5 32163  ax-c4 32164  ax-c7 32165  ax-c11 32167  ax-c9 32170
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  axc16g-o  32213
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