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Theorem ax11b 1938
Description: A bidirectional version of ax11o 1937. (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
ax11b  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  <->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem ax11b
StepHypRef Expression
1 ax11o 1937 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
21imp 420 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) )
3 ax4 1717 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
43com12 29 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
54adantl 454 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( A. x ( x  =  y  ->  ph )  ->  ph )
)
62, 5impbid 185 1  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  <->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530  df-nf 1533
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