Theorem hboprab1 3984

Description: The abstraction variables in an operation class abstraction are not free.

Assertion

Ref	Expression
hboprab1	|- (w e. {<.<.x, y>., z>. | ph} -> A.x w e. {<.<.x, y>., z>. | ph})

Distinct variable groups:   x,y,z   x,w

Proof of Theorem hboprab1

Step	Hyp	Ref	Expression
1		df-oprab 3957	. 2 |- {<.<.x, y>., z>. | ph} = {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
2		hbe1 1014	. . 3 |- (E.xE.yE.z(v = <.<.x, y>., z>. /\ ph) -> A.xE.xE.yE.z(v = <.<.x, y>., z>. /\ ph))
3	2	hbab 1465	. 2 |- (w e. {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)} -> A.x w e. {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)})
4	1, 3	hbxfr 1560	1 |- (w e. {<.<.x, y>., z>. | ph} -> A.x w e. {<.<.x, y>., z>. | ph})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  {cab 1461  <.cop 2407  {copab2 3955

This theorem is referenced by:  elrnoprabg 4114  mapxpen 4481

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-oprab 3957

Copyright terms: Public domain