Theorem hbopab2 2820

Description: The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free.

Assertion

Ref	Expression
hbopab2	|- (z e. {<.x, y>. | ph} -> A.y z e. {<.x, y>. | ph})

Distinct variable group:   y,z

Proof of Theorem hbopab2

Step	Hyp	Ref	Expression
1		df-opab 2672	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
2		hbe1 1018	. . . 4 |- (E.y(w = <.x, y>. /\ ph) -> A.yE.y(w = <.x, y>. /\ ph))
3	2	hbex 1008	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) -> A.yE.xE.y(w = <.x, y>. /\ ph))
4	3	hbab 1470	. 2 |- (z e. {w | E.xE.y(w = <.x, y>. /\ ph)} -> A.y z e. {w | E.xE.y(w = <.x, y>. /\ ph)})
5	1, 4	hbxfr 1566	1 |- (z e. {<.x, y>. | ph} -> A.y z e. {<.x, y>. | ph})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  <.cop 2415  {copab 2671

This theorem is referenced by:  opabsb 2821  ssopab2 2828  dmopab 3326  rnopab 3359  cnvopab 3451  funopab 3554  zfrep6 3620  fnopabg 3621  fvopab2 3797  fvopab5 3799  fopab2 3829  abrexexlem2 3865  dom2d 4410  xpmapenlem1 4502  xpmapenlem4 4505

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-opab 2672

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