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Theorem ax12b 2141
Description: A bidirectional version of axc15 2140. (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
ax12b  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  <->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem ax12b
StepHypRef Expression
1 axc15 2140 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
21imp 430 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) )
3 sp 1910 . . . 4  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
43com12 32 . . 3  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
54adantl 467 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( A. x ( x  =  y  ->  ph )  ->  ph )
)
62, 5impbid 193 1  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ph  <->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by: (None)
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