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

Theorem sbieh 1714
Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1715 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)
Hypotheses
Ref Expression
sbieh.1  |-  ( ps 
->  A. x ps )
sbieh.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbieh  |-  ( [ y  /  x ] ph 
<->  ps )

Proof of Theorem sbieh
StepHypRef Expression
1 id 19 . 2  |-  ( ph  ->  ph )
21hbth 1393 . . 3  |-  ( (
ph  ->  ph )  ->  A. x
( ph  ->  ph )
)
3 sbieh.1 . . . 4  |-  ( ps 
->  A. x ps )
43a1i 9 . . 3  |-  ( (
ph  ->  ph )  ->  ( ps  ->  A. x ps )
)
5 sbieh.2 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
65a1i 9 . . 3  |-  ( (
ph  ->  ph )  ->  (
x  =  y  -> 
( ph  <->  ps ) ) )
72, 4, 6sbiedh 1711 . 2  |-  ( (
ph  ->  ph )  ->  ( [ y  /  x ] ph  <->  ps ) )
81, 7ax-mp 7 1  |-  ( [ y  /  x ] ph 
<->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   [wsb 1686
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sbie  1715  sbco2vlem  1862  equsb3lem  1866  sbco2yz  1879  elsb3  1894  elsb4  1895  dvelimf  1933
  Copyright terms: Public domain W3C validator