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Theorem ax11v 2174
Description: This is a version of ax-11o 2220 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2079 for the rederivation of ax-11o 2220 from this theorem. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax11v  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax11v
StepHypRef Expression
1 ax-1 5 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ph ) )
2 ax16 2051 . . . 4  |-  ( A. x  x  =  y  ->  ( ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  ph ) ) )
31, 2syl5 31 . . 3  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x
( x  =  y  ->  ph ) ) )
43a1d 24 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
5 ax11o 2082 . 2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
64, 5pm2.61i 159 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550
This theorem is referenced by:  sb56  2176  exsb  2209  rexsb  27924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555
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