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

Theorem r19.2m 3524
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1649). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
Assertion
Ref Expression
r19.2m  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem r19.2m
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2250 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
21cbvexv 1930 . . 3  |-  ( E. x  x  e.  A  <->  E. z  z  e.  A
)
3 eleq1w 2250 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
43cbvexv 1930 . . 3  |-  ( E. z  z  e.  A  <->  E. y  y  e.  A
)
52, 4bitri 184 . 2  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
6 df-ral 2473 . . . . 5  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
7 exintr 1645 . . . . 5  |-  ( A. x ( x  e.  A  ->  ph )  -> 
( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ph ) ) )
86, 7sylbi 121 . . . 4  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ph ) ) )
9 df-rex 2474 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
108, 9imbitrrdi 162 . . 3  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  E. x  e.  A  ph ) )
1110impcom 125 . 2  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
125, 11sylanbr 285 1  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1362   E.wex 1503    e. wcel 2160   A.wral 2468   E.wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-clel 2185  df-ral 2473  df-rex 2474
This theorem is referenced by:  intssunim  3881  riinm  3974  iinexgm  4172  xpiindim  4782  cnviinm  5188  eusvobj2  5881  iinerm  6632  suplocexprlemml  7744  rexfiuz  11029  r19.2uz  11033  climuni  11332  pc2dvds  12361  issubg4m  13129  cncnp2m  14183
  Copyright terms: Public domain W3C validator