Theorem hboprab1 3993

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 3966	. 2 |- {<.<.x, y>., z>. | ph} = {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
2		hbe1 1016	. . 3 |- (E.xE.yE.z(v = <.<.x, y>., z>. /\ ph) -> A.xE.xE.yE.z(v = <.<.x, y>., z>. /\ ph))
3	2	hbab 1467	. 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 1563	1 |- (w e. {<.<.x, y>., z>. | ph} -> A.x w e. {<.<.x, y>., z>. | ph})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  <.cop 2411  {copab2 3964

This theorem is referenced by:  elrnoprabg 4124  mapxpen 4495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-oprab 3966

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