ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbco2vlem Unicode version

Theorem sbco2vlem 1956
Description: This is a version of sbco2 1977 where  z is distinct from 
x and from  y. It is a lemma on the way to proving sbco2v 1960 which only requires that  z and  x be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) Remove one disjoint variable condition. (Revised by Jim Kingdon, 3-Feb-2018.)
Hypothesis
Ref Expression
sbco2vlem.1  |-  ( ph  ->  A. z ph )
Assertion
Ref Expression
sbco2vlem  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbco2vlem
StepHypRef Expression
1 sbco2vlem.1 . . 3  |-  ( ph  ->  A. z ph )
21hbsbv 1953 . 2  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
3 sbequ 1851 . 2  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
42, 3sbieh 1801 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1362   [wsb 1773
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  sbco2vh  1957
  Copyright terms: Public domain W3C validator