ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rgenm Unicode version

Theorem rgenm 3496
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 1476 . . . . 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 1502 . . . 4  |-  ( E. x  x  e.  A  ->  A. x ( x  e.  A  ->  ph )
)
5 19.38 1656 . . . 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 1450 . . 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 2440 . 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 1333   E.wex 1472    e. wcel 2128   A.wral 2435
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator