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Theorem aev 1766
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1768. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
aev  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Distinct variable group:    x, y

Proof of Theorem aev
Dummy variables  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbae 1679 . 2  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
2 hbae 1679 . . . 4  |-  ( A. x  x  =  y  ->  A. f A. x  x  =  y )
3 ax-8 1465 . . . . 5  |-  ( x  =  f  ->  (
x  =  y  -> 
f  =  y ) )
43spimv 1765 . . . 4  |-  ( A. x  x  =  y  ->  f  =  y )
52, 4alrimih 1428 . . 3  |-  ( A. x  x  =  y  ->  A. f  f  =  y )
6 ax-8 1465 . . . . . . . 8  |-  ( y  =  u  ->  (
y  =  f  ->  u  =  f )
)
7 equcomi 1663 . . . . . . . 8  |-  ( u  =  f  ->  f  =  u )
86, 7syl6 33 . . . . . . 7  |-  ( y  =  u  ->  (
y  =  f  -> 
f  =  u ) )
98spimv 1765 . . . . . 6  |-  ( A. y  y  =  f  ->  f  =  u )
109alequcoms 1479 . . . . 5  |-  ( A. f  f  =  y  ->  f  =  u )
1110a5i 1505 . . . 4  |-  ( A. f  f  =  y  ->  A. f  f  =  u )
12 hbae 1679 . . . . 5  |-  ( A. f  f  =  u  ->  A. v A. f 
f  =  u )
13 ax-8 1465 . . . . . 6  |-  ( f  =  v  ->  (
f  =  u  -> 
v  =  u ) )
1413spimv 1765 . . . . 5  |-  ( A. f  f  =  u  ->  v  =  u )
1512, 14alrimih 1428 . . . 4  |-  ( A. f  f  =  u  ->  A. v  v  =  u )
16 alequcom 1478 . . . 4  |-  ( A. v  v  =  u  ->  A. u  u  =  v )
1711, 15, 163syl 17 . . 3  |-  ( A. f  f  =  y  ->  A. u  u  =  v )
18 ax-8 1465 . . . 4  |-  ( u  =  w  ->  (
u  =  v  ->  w  =  v )
)
1918spimv 1765 . . 3  |-  ( A. u  u  =  v  ->  w  =  v )
205, 17, 193syl 17 . 2  |-  ( A. x  x  =  y  ->  w  =  v )
211, 20alrimih 1428 1  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  ax16  1767  a16g  1818
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