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Theorem ax11b 1935
Description: A bidirectional version of ax11o 1934. (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 1934 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
21imp 418 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) )
3 sp 1716 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
43com12 27 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
54adantl 452 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( A. x ( x  =  y  ->  ph )  ->  ph )
)
62, 5impbid 183 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    <-> wb 176    /\ wa 358   A.wal 1527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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