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Theorem rgenm 3433
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 1455 . . . . 5  |-  F/ x E. x  x  e.  A
2 rgenm.1 . . . . . 6  |-  ( ( E. x  x  e.  A  /\  x  e.  A )  ->  ph )
32ex 114 . . . . 5  |-  ( E. x  x  e.  A  ->  ( x  e.  A  ->  ph ) )
41, 3alrimi 1485 . . . 4  |-  ( E. x  x  e.  A  ->  A. x ( x  e.  A  ->  ph )
)
5 19.38 1637 . . . 4  |-  ( ( E. x  x  e.  A  ->  A. x
( x  e.  A  ->  ph ) )  ->  A. x ( x  e.  A  ->  ( x  e.  A  ->  ph )
) )
64, 5ax-mp 5 . . 3  |-  A. x
( x  e.  A  ->  ( x  e.  A  ->  ph ) )
7 pm5.4 248 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  A  ->  ph ) )  <->  ( x  e.  A  ->  ph )
)
87albii 1429 . . 3  |-  ( A. x ( x  e.  A  ->  ( x  e.  A  ->  ph )
)  <->  A. x ( x  e.  A  ->  ph )
)
96, 8mpbi 144 . 2  |-  A. x
( x  e.  A  ->  ph )
10 df-ral 2396 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
119, 10mpbir 145 1  |-  A. x  e.  A  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312   E.wex 1451    e. wcel 1463   A.wral 2391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-ral 2396
This theorem is referenced by: (None)
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