ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rmo4f Unicode version

Theorem rmo4f 2910
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
rmo4f.1  |-  F/_ x A
rmo4f.2  |-  F/_ y A
rmo4f.3  |-  F/ x ps
rmo4f.4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rmo4f  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    A( x, y)

Proof of Theorem rmo4f
StepHypRef Expression
1 rmo4f.1 . . 3  |-  F/_ x A
2 rmo4f.2 . . 3  |-  F/_ y A
3 nfv 1508 . . 3  |-  F/ y
ph
41, 2, 3rmo3f 2909 . 2  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
5 rmo4f.3 . . . . . 6  |-  F/ x ps
6 rmo4f.4 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
75, 6sbie 1771 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
87anbi2i 453 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  ps )
)
98imbi1i 237 . . 3  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  y )
)
1092ralbii 2465 . 2  |-  ( A. x  e.  A  A. y  e.  A  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
114, 10bitri 183 1  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   F/wnf 1440   [wsb 1742   F/_wnfc 2286   A.wral 2435   E*wrmo 2438
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rmo 2443
This theorem is referenced by:  disjxp1  6184
  Copyright terms: Public domain W3C validator