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Theorem rgenm 3371
Description: Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
Hypothesis
Ref Expression
rgenm.1  |-  ( ( E. x  x  e.  A  /\  x  e.  A )  ->  ph )
Assertion
Ref Expression
rgenm  |-  A. x  e.  A  ph
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rgenm
StepHypRef Expression
1 nfe1 1428 . . . . 5  |-  F/ x E. x  x  e.  A
2 rgenm.1 . . . . . 6  |-  ( ( E. x  x  e.  A  /\  x  e.  A )  ->  ph )
32ex 113 . . . . 5  |-  ( E. x  x  e.  A  ->  ( x  e.  A  ->  ph ) )
41, 3alrimi 1458 . . . 4  |-  ( E. x  x  e.  A  ->  A. x ( x  e.  A  ->  ph )
)
5 19.38 1609 . . . 4  |-  ( ( E. x  x  e.  A  ->  A. x
( x  e.  A  ->  ph ) )  ->  A. x ( x  e.  A  ->  ( x  e.  A  ->  ph )
) )
64, 5ax-mp 7 . . 3  |-  A. x
( x  e.  A  ->  ( x  e.  A  ->  ph ) )
7 pm5.4 247 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  A  ->  ph ) )  <->  ( x  e.  A  ->  ph )
)
87albii 1402 . . 3  |-  ( A. x ( x  e.  A  ->  ( x  e.  A  ->  ph )
)  <->  A. x ( x  e.  A  ->  ph )
)
96, 8mpbi 143 . 2  |-  A. x
( x  e.  A  ->  ph )
10 df-ral 2360 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
119, 10mpbir 144 1  |-  A. x  e.  A  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1285   E.wex 1424    e. wcel 1436   A.wral 2355
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-ral 2360
This theorem is referenced by: (None)
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