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

Theorem hbab 2047
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
Hypothesis
Ref Expression
hbab.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
hbab  |-  ( z  e.  { y  | 
ph }  ->  A. x  z  e.  { y  |  ph } )
Distinct variable group:    x, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem hbab
StepHypRef Expression
1 df-clab 2043 . 2  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
2 hbab.1 . . 3  |-  ( ph  ->  A. x ph )
32hbsb 1839 . 2  |-  ( [ z  /  y ]
ph  ->  A. x [ z  /  y ] ph )
41, 3hbxfrbi 1377 1  |-  ( z  e.  { y  | 
ph }  ->  A. x  z  e.  { y  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    e. wcel 1409   [wsb 1661   {cab 2042
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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043
This theorem is referenced by:  nfsab  2048
  Copyright terms: Public domain W3C validator