ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.2m Unicode version

Theorem r19.2m 3365
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1574). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
Assertion
Ref Expression
r19.2m  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem r19.2m
StepHypRef Expression
1 df-ral 2364 . . . 4  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
2 exintr 1570 . . . 4  |-  ( A. x ( x  e.  A  ->  ph )  -> 
( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ph ) ) )
31, 2sylbi 119 . . 3  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ph ) ) )
4 df-rex 2365 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
53, 4syl6ibr 160 . 2  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  E. x  e.  A  ph ) )
65impcom 123 1  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287   E.wex 1426    e. wcel 1438   A.wral 2359   E.wrex 2360
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-ial 1472
This theorem depends on definitions:  df-bi 115  df-ral 2364  df-rex 2365
This theorem is referenced by:  intssunim  3705  riinm  3797  trintssmOLD  3945  iinexgm  3982  xpiindim  4561  cnviinm  4959  eusvobj2  5620  iinerm  6344  rexfiuz  10387  r19.2uz  10391  climuni  10645
  Copyright terms: Public domain W3C validator