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Theorem ax11b 1836
Description: A bidirectional version of ax-11o 1833. (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 1832 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
21imp 124 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) )
3 ax-4 1520 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
43com12 30 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
54adantl 277 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( A. x ( x  =  y  ->  ph )  ->  ph )
)
62, 5impbid 129 1  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  <->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773
This theorem is referenced by: (None)
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