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Theorem aev-o 2154
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2116. Version of aev 1963 using ax-10o 2111. (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 2125 . 2  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
2 hbae-o 2125 . . . 4  |-  ( A. x  x  =  y  ->  A. t A. x  x  =  y )
3 ax-8 1666 . . . . 5  |-  ( x  =  t  ->  (
x  =  y  -> 
t  =  y ) )
43spimv 1962 . . . 4  |-  ( A. x  x  =  y  ->  t  =  y )
52, 4alrimih 1556 . . 3  |-  ( A. x  x  =  y  ->  A. t  t  =  y )
6 ax-8 1666 . . . . . . . 8  |-  ( y  =  u  ->  (
y  =  t  ->  u  =  t )
)
7 equcomi 1670 . . . . . . . 8  |-  ( u  =  t  ->  t  =  u )
86, 7syl6 29 . . . . . . 7  |-  ( y  =  u  ->  (
y  =  t  -> 
t  =  u ) )
98spimv 1962 . . . . . 6  |-  ( A. y  y  =  t  ->  t  =  u )
109aecoms-o 2124 . . . . 5  |-  ( A. t  t  =  y  ->  t  =  u )
1110a5i-o 2122 . . . 4  |-  ( A. t  t  =  y  ->  A. t  t  =  u )
12 hbae-o 2125 . . . . 5  |-  ( A. t  t  =  u  ->  A. v A. t 
t  =  u )
13 ax-8 1666 . . . . . 6  |-  ( t  =  v  ->  (
t  =  u  -> 
v  =  u ) )
1413spimv 1962 . . . . 5  |-  ( A. t  t  =  u  ->  v  =  u )
1512, 14alrimih 1556 . . . 4  |-  ( A. t  t  =  u  ->  A. v  v  =  u )
16 aecom-o 2123 . . . 4  |-  ( A. v  v  =  u  ->  A. u  u  =  v )
1711, 15, 163syl 18 . . 3  |-  ( A. t  t  =  y  ->  A. u  u  =  v )
18 ax-8 1666 . . . 4  |-  ( u  =  w  ->  (
u  =  v  ->  w  =  v )
)
1918spimv 1962 . . 3  |-  ( A. u  u  =  v  ->  w  =  v )
205, 17, 193syl 18 . 2  |-  ( A. x  x  =  y  ->  w  =  v )
211, 20alrimih 1556 1  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1531
This theorem is referenced by:  a16g-o  2158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-4 2107  ax-5o 2108  ax-6o 2109  ax-10o 2111  ax-12o 2114
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536
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