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Theorem ax9o 1633
Description: An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
ax9o  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )

Proof of Theorem ax9o
StepHypRef Expression
1 a9e 1631 . 2  |-  E. x  x  =  y
2 19.29r 1557 . . 3  |-  ( ( E. x  x  =  y  /\  A. x
( x  =  y  ->  A. x ph )
)  ->  E. x
( x  =  y  /\  ( x  =  y  ->  A. x ph ) ) )
3 hba1 1478 . . . . 5  |-  ( A. x ph  ->  A. x A. x ph )
4 pm3.35 339 . . . . 5  |-  ( ( x  =  y  /\  ( x  =  y  ->  A. x ph )
)  ->  A. x ph )
53, 4exlimih 1529 . . . 4  |-  ( E. x ( x  =  y  /\  ( x  =  y  ->  A. x ph ) )  ->  A. x ph )
6 ax-4 1445 . . . 4  |-  ( A. x ph  ->  ph )
75, 6syl 14 . . 3  |-  ( E. x ( x  =  y  /\  ( x  =  y  ->  A. x ph ) )  ->  ph )
82, 7syl 14 . 2  |-  ( ( E. x  x  =  y  /\  A. x
( x  =  y  ->  A. x ph )
)  ->  ph )
91, 8mpan 415 1  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289   E.wex 1426
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-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  equsalh  1661  spimth  1670  spimh  1672
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