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Theorem ax11b 1748
Description: A bidirectional version of ax-11o 1745. (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 1744 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
21imp 122 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) )
3 ax-4 1441 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
43com12 30 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
54adantl 271 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( A. x ( x  =  y  ->  ph )  ->  ph )
)
62, 5impbid 127 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 102    <-> wb 103   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by: (None)
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