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

Theorem rmobida 2513
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmobida.1  |-  F/ x ph
rmobida.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rmobida  |-  ( ph  ->  ( E* x  e.  A  ps  <->  E* x  e.  A  ch )
)

Proof of Theorem rmobida
StepHypRef Expression
1 rmobida.1 . . 3  |-  F/ x ph
2 rmobida.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32pm5.32da 433 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3mobid 1951 . 2  |-  ( ph  ->  ( E* x ( x  e.  A  /\  ps )  <->  E* x ( x  e.  A  /\  ch ) ) )
5 df-rmo 2331 . 2  |-  ( E* x  e.  A  ps  <->  E* x ( x  e.  A  /\  ps )
)
6 df-rmo 2331 . 2  |-  ( E* x  e.  A  ch  <->  E* x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 216 1  |-  ( ph  ->  ( E* x  e.  A  ps  <->  E* x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   F/wnf 1365    e. wcel 1409   E*wmo 1917   E*wrmo 2326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-eu 1919  df-mo 1920  df-rmo 2331
This theorem is referenced by:  rmobidva  2514
  Copyright terms: Public domain W3C validator