Theorem cbvoprab12 3028

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution.

Hypotheses

Ref	Expression
cbvoprab12.1	|- (ph -> A.wph)
cbvoprab12.2	|- (ph -> A.vph)
cbvoprab12.3	|- (ps -> A.xps)
cbvoprab12.4	|- (ps -> A.yps)
cbvoprab12.5	|- ((x = w /\ y = v) -> (ph <-> ps))

Assertion

Ref	Expression
cbvoprab12	|- {<.<.x, y>., z>. | ph} = {<.<.w, v>., z>. | ps}

Distinct variable group(s):   x,y,z,w,v

Proof of Theorem cbvoprab12

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . . 6 |- (u = <.<.x, y>., z>. -> A.w u = <.<.x, y>., z>.)
2		cbvoprab12.1	. . . . . 6 |- (ph -> A.wph)
3	1, 2	hban 704	. . . . 5 |- ((u = <.<.x, y>., z>. /\ ph) -> A.w(u = <.<.x, y>., z>. /\ ph))
4	3	hbex 701	. . . 4 |- (E.z(u = <.<.x, y>., z>. /\ ph) -> A.wE.z(u = <.<.x, y>., z>. /\ ph))
5		ax-17 925	. . . . . 6 |- (u = <.<.x, y>., z>. -> A.v u = <.<.x, y>., z>.)
6		cbvoprab12.2	. . . . . 6 |- (ph -> A.vph)
7	5, 6	hban 704	. . . . 5 |- ((u = <.<.x, y>., z>. /\ ph) -> A.v(u = <.<.x, y>., z>. /\ ph))
8	7	hbex 701	. . . 4 |- (E.z(u = <.<.x, y>., z>. /\ ph) -> A.vE.z(u = <.<.x, y>., z>. /\ ph))
9		ax-17 925	. . . . . 6 |- (u = <.<.w, v>., z>. -> A.x u = <.<.w, v>., z>.)
10		cbvoprab12.3	. . . . . 6 |- (ps -> A.xps)
11	9, 10	hban 704	. . . . 5 |- ((u = <.<.w, v>., z>. /\ ps) -> A.x(u = <.<.w, v>., z>. /\ ps))
12	11	hbex 701	. . . 4 |- (E.z(u = <.<.w, v>., z>. /\ ps) -> A.xE.z(u = <.<.w, v>., z>. /\ ps))
13		ax-17 925	. . . . . 6 |- (u = <.<.w, v>., z>. -> A.y u = <.<.w, v>., z>.)
14		cbvoprab12.4	. . . . . 6 |- (ps -> A.yps)
15	13, 14	hban 704	. . . . 5 |- ((u = <.<.w, v>., z>. /\ ps) -> A.y(u = <.<.w, v>., z>. /\ ps))
16	15	hbex 701	. . . 4 |- (E.z(u = <.<.w, v>., z>. /\ ps) -> A.yE.z(u = <.<.w, v>., z>. /\ ps))
17		opeq12 1878	. . . . . . . 8 |- ((x = w /\ y = v) -> <.x, y>. = <.w, v>.)
18		opeq1 1876	. . . . . . . 8 |- (<.x, y>. = <.w, v>. -> <.<.x, y>., z>. = <.<.w, v>., z>.)
19	17, 18	syl 12	. . . . . . 7 |- ((x = w /\ y = v) -> <.<.x, y>., z>. = <.<.w, v>., z>.)
20	19	cleq2d 1112	. . . . . 6 |- ((x = w /\ y = v) -> (u = <.<.x, y>., z>. <-> u = <.<.w, v>., z>.))
21		cbvoprab12.5	. . . . . 6 |- ((x = w /\ y = v) -> (ph <-> ps))
22	20, 21	anbi12d 476	. . . . 5 |- ((x = w /\ y = v) -> ((u = <.<.x, y>., z>. /\ ph) <-> (u = <.<.w, v>., z>. /\ ps)))
23	22	biexdv 936	. . . 4 |- ((x = w /\ y = v) -> (E.z(u = <.<.x, y>., z>. /\ ph) <-> E.z(u = <.<.w, v>., z>. /\ ps)))
24	4, 8, 12, 16, 23	cbvex2 975	. . 3 |- (E.xE.yE.z(u = <.<.x, y>., z>. /\ ph) <-> E.wE.vE.z(u = <.<.w, v>., z>. /\ ps))
25	24	biabi 1181	. 2 |- {u | E.xE.yE.z(u = <.<.x, y>., z>. /\ ph)} = {u | E.wE.vE.z(u = <.<.w, v>., z>. /\ ps)}
26		df-oprab 3004	. 2 |- {<.<.x, y>., z>. | ph} = {u | E.xE.yE.z(u = <.<.x, y>., z>. /\ ph)}
27		df-oprab 3004	. 2 |- {<.<.w, v>., z>. | ps} = {u | E.wE.vE.z(u = <.<.w, v>., z>. /\ ps)}
28	25, 26, 27	3eqtr4 1126	1 |- {<.<.x, y>., z>. | ph} = {<.<.w, v>., z>. | ps}

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  {cab 1090   = wceq 1091  <.cop 1810  {copab2 3002

This theorem is referenced by:  cbvoprab12v 3029  oprabval4g 3053

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-oprab 3004

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