Theorem cbvopab1 2106

Description: Change first bound variable in an ordered pair abstraction, using explicit substitution.

Hypotheses

Ref	Expression
cbvopab1.1	|- (ph -> A.zph)
cbvopab1.2	|- (ps -> A.xps)
cbvopab1.3	|- (x = z -> (ph <-> ps))

Assertion

Ref	Expression
cbvopab1	|- {<.x, y>. | ph} = {<.z, y>. | ps}

Distinct variable group(s):   x,y,z

Proof of Theorem cbvopab1

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . . 6 |- (w = <.x, y>. -> A.z w = <.x, y>.)
2		cbvopab1.1	. . . . . 6 |- (ph -> A.zph)
3	1, 2	hban 704	. . . . 5 |- ((w = <.x, y>. /\ ph) -> A.z(w = <.x, y>. /\ ph))
4	3	hbex 701	. . . 4 |- (E.y(w = <.x, y>. /\ ph) -> A.zE.y(w = <.x, y>. /\ ph))
5		ax-17 925	. . . . . 6 |- (w = <.z, y>. -> A.x w = <.z, y>.)
6		cbvopab1.2	. . . . . 6 |- (ps -> A.xps)
7	5, 6	hban 704	. . . . 5 |- ((w = <.z, y>. /\ ps) -> A.x(w = <.z, y>. /\ ps))
8	7	hbex 701	. . . 4 |- (E.y(w = <.z, y>. /\ ps) -> A.xE.y(w = <.z, y>. /\ ps))
9		opeq1 1876	. . . . . . 7 |- (x = z -> <.x, y>. = <.z, y>.)
10	9	cleq2d 1112	. . . . . 6 |- (x = z -> (w = <.x, y>. <-> w = <.z, y>.))
11		cbvopab1.3	. . . . . 6 |- (x = z -> (ph <-> ps))
12	10, 11	anbi12d 476	. . . . 5 |- (x = z -> ((w = <.x, y>. /\ ph) <-> (w = <.z, y>. /\ ps)))
13	12	biexdv 936	. . . 4 |- (x = z -> (E.y(w = <.x, y>. /\ ph) <-> E.y(w = <.z, y>. /\ ps)))
14	4, 8, 13	cbvex 849	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.zE.y(w = <.z, y>. /\ ps))
15	14	biabi 1181	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.zE.y(w = <.z, y>. /\ ps)}
16		df-opab 2098	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
17		df-opab 2098	. 2 |- {<.z, y>. | ps} = {w | E.zE.y(w = <.z, y>. /\ ps)}
18	15, 16, 17	3eqtr4 1126	1 |- {<.x, y>. | ph} = {<.z, y>. | ps}

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  {cab 1090   = wceq 1091  <.cop 1810  {copab 2055

This theorem is referenced by:  seqlem1 4662

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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