Theorem hboprab 5563

Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by set.mm contributors, 22-Aug-2013.)

Hypothesis

Ref	Expression
hboprab.1	|- (ph -> A.wph)

Assertion

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

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

Allowed substitution hints:   ph(x,y,z,w,u)

Proof of Theorem hboprab

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 5529	. 2 |- {<.<.x, y>., z>. | ph} = {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
2		ax-17 1616	. . . . . . 7 |- (v = <.<.x, y>., z>. -> A.w v = <.<.x, y>., z>.)
3		hboprab.1	. . . . . . 7 |- (ph -> A.wph)
4	2, 3	hban 1828	. . . . . 6 |- ((v = <.<.x, y>., z>. /\ ph) -> A.w(v = <.<.x, y>., z>. /\ ph))
5	4	hbex 1841	. . . . 5 |- (E.z(v = <.<.x, y>., z>. /\ ph) -> A.wE.z(v = <.<.x, y>., z>. /\ ph))
6	5	hbex 1841	. . . 4 |- (E.yE.z(v = <.<.x, y>., z>. /\ ph) -> A.wE.yE.z(v = <.<.x, y>., z>. /\ ph))
7	6	hbex 1841	. . 3 |- (E.xE.yE.z(v = <.<.x, y>., z>. /\ ph) -> A.wE.xE.yE.z(v = <.<.x, y>., z>. /\ ph))
8	7	hbab 2344	. 2 |- (u e. {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)} -> A.w u e. {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)})
9	1, 8	hbxfreq 2457	1 |- (u e. {<.<.x, y>., z>. | ph} -> A.w u e. {<.<.x, y>., z>. | ph})

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  <.cop 4562  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-oprab 5529

This theorem is referenced by: (None)