Theorem cbvoprab12 3998

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:   x,y,z,w,v

Proof of Theorem cbvoprab12

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . . 6 |- (u = <.<.x, y>., z>. -> A.w u = <.<.x, y>., z>.)
2		cbvoprab12.1	. . . . . 6 |- (ph -> A.wph)
3	1, 2	hban 1009	. . . . 5 |- ((u = <.<.x, y>., z>. /\ ph) -> A.w(u = <.<.x, y>., z>. /\ ph))
4	3	hbex 1006	. . . 4 |- (E.z(u = <.<.x, y>., z>. /\ ph) -> A.wE.z(u = <.<.x, y>., z>. /\ ph))
5		ax-17 971	. . . . . 6 |- (u = <.<.x, y>., z>. -> A.v u = <.<.x, y>., z>.)
6		cbvoprab12.2	. . . . . 6 |- (ph -> A.vph)
7	5, 6	hban 1009	. . . . 5 |- ((u = <.<.x, y>., z>. /\ ph) -> A.v(u = <.<.x, y>., z>. /\ ph))
8	7	hbex 1006	. . . 4 |- (E.z(u = <.<.x, y>., z>. /\ ph) -> A.vE.z(u = <.<.x, y>., z>. /\ ph))
9		ax-17 971	. . . . . 6 |- (u = <.<.w, v>., z>. -> A.x u = <.<.w, v>., z>.)
10		cbvoprab12.3	. . . . . 6 |- (ps -> A.xps)
11	9, 10	hban 1009	. . . . 5 |- ((u = <.<.w, v>., z>. /\ ps) -> A.x(u = <.<.w, v>., z>. /\ ps))
12	11	hbex 1006	. . . 4 |- (E.z(u = <.<.w, v>., z>. /\ ps) -> A.xE.z(u = <.<.w, v>., z>. /\ ps))
13		ax-17 971	. . . . . 6 |- (u = <.<.w, v>., z>. -> A.y u = <.<.w, v>., z>.)
14		cbvoprab12.4	. . . . . 6 |- (ps -> A.yps)
15	13, 14	hban 1009	. . . . 5 |- ((u = <.<.w, v>., z>. /\ ps) -> A.y(u = <.<.w, v>., z>. /\ ps))
16	15	hbex 1006	. . . 4 |- (E.z(u = <.<.w, v>., z>. /\ ps) -> A.yE.z(u = <.<.w, v>., z>. /\ ps))
17		opeq12 2489	. . . . . . . 8 |- ((x = w /\ y = v) -> <.x, y>. = <.w, v>.)
18	17	opeq1d 2493	. . . . . . 7 |- ((x = w /\ y = v) -> <.<.x, y>., z>. = <.<.w, v>., z>.)
19	18	eqeq2d 1486	. . . . . 6 |- ((x = w /\ y = v) -> (u = <.<.x, y>., z>. <-> u = <.<.w, v>., z>.))
20		cbvoprab12.5	. . . . . 6 |- ((x = w /\ y = v) -> (ph <-> ps))
21	19, 20	anbi12d 628	. . . . 5 |- ((x = w /\ y = v) -> ((u = <.<.x, y>., z>. /\ ph) <-> (u = <.<.w, v>., z>. /\ ps)))
22	21	exbidv 1279	. . . 4 |- ((x = w /\ y = v) -> (E.z(u = <.<.x, y>., z>. /\ ph) <-> E.z(u = <.<.w, v>., z>. /\ ps)))
23	4, 8, 12, 16, 22	cbvex2 1317	. . 3 |- (E.xE.yE.z(u = <.<.x, y>., z>. /\ ph) <-> E.wE.vE.z(u = <.<.w, v>., z>. /\ ps))
24	23	abbii 1575	. 2 |- {u | E.xE.yE.z(u = <.<.x, y>., z>. /\ ph)} = {u | E.wE.vE.z(u = <.<.w, v>., z>. /\ ps)}
25		df-oprab 3966	. 2 |- {<.<.x, y>., z>. | ph} = {u | E.xE.yE.z(u = <.<.x, y>., z>. /\ ph)}
26		df-oprab 3966	. 2 |- {<.<.w, v>., z>. | ps} = {u | E.wE.vE.z(u = <.<.w, v>., z>. /\ ps)}
27	24, 25, 26	3eqtr4 1505	1 |- {<.<.x, y>., z>. | ph} = {<.<.w, v>., z>. | ps}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956  E.wex 980  {cab 1463  <.cop 2411  {copab2 3964

This theorem is referenced by:  cbvoprab12v 3999  oprabval2gf 4026  oprabval4g 4031

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-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-oprab 3966

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