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Theorem a12lem2 29753
Description: Proof of second hypothesis of a12study 29754. (Contributed by NM, 15-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a12lem2  |-  ( A. z ( z  =  x  ->  -.  z  =  y )  ->  -.  x  =  y
)

Proof of Theorem a12lem2
StepHypRef Expression
1 equcom 1665 . . . . . 6  |-  ( z  =  x  <->  x  =  z )
21imbi1i 315 . . . . 5  |-  ( ( z  =  x  ->  -.  z  =  y
)  <->  ( x  =  z  ->  -.  z  =  y ) )
3 imnan 411 . . . . 5  |-  ( ( x  =  z  ->  -.  z  =  y
)  <->  -.  ( x  =  z  /\  z  =  y ) )
42, 3bitri 240 . . . 4  |-  ( ( z  =  x  ->  -.  z  =  y
)  <->  -.  ( x  =  z  /\  z  =  y ) )
54albii 1556 . . 3  |-  ( A. z ( z  =  x  ->  -.  z  =  y )  <->  A. z  -.  ( x  =  z  /\  z  =  y ) )
6 alnex 1533 . . 3  |-  ( A. z  -.  ( x  =  z  /\  z  =  y )  <->  -.  E. z
( x  =  z  /\  z  =  y ) )
75, 6bitri 240 . 2  |-  ( A. z ( z  =  x  ->  -.  z  =  y )  <->  -.  E. z
( x  =  z  /\  z  =  y ) )
8 equvini 1940 . . 3  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
98con3i 127 . 2  |-  ( -. 
E. z ( x  =  z  /\  z  =  y )  ->  -.  x  =  y
)
107, 9sylbi 187 1  |-  ( A. z ( z  =  x  ->  -.  z  =  y )  ->  -.  x  =  y
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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