Theorem cbvopab 2672

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution.

Hypotheses

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

Assertion

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

Distinct variable group:   x,y,z,w

Proof of Theorem cbvopab

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . 5 |- (v = <.x, y>. -> A.z v = <.x, y>.)
2		cbvopab.1	. . . . 5 |- (ph -> A.zph)
3	1, 2	hban 1009	. . . 4 |- ((v = <.x, y>. /\ ph) -> A.z(v = <.x, y>. /\ ph))
4		ax-17 971	. . . . 5 |- (v = <.x, y>. -> A.w v = <.x, y>.)
5		cbvopab.2	. . . . 5 |- (ph -> A.wph)
6	4, 5	hban 1009	. . . 4 |- ((v = <.x, y>. /\ ph) -> A.w(v = <.x, y>. /\ ph))
7		ax-17 971	. . . . 5 |- (v = <.z, w>. -> A.x v = <.z, w>.)
8		cbvopab.3	. . . . 5 |- (ps -> A.xps)
9	7, 8	hban 1009	. . . 4 |- ((v = <.z, w>. /\ ps) -> A.x(v = <.z, w>. /\ ps))
10		ax-17 971	. . . . 5 |- (v = <.z, w>. -> A.y v = <.z, w>.)
11		cbvopab.4	. . . . 5 |- (ps -> A.yps)
12	10, 11	hban 1009	. . . 4 |- ((v = <.z, w>. /\ ps) -> A.y(v = <.z, w>. /\ ps))
13		opeq12 2489	. . . . . 6 |- ((x = z /\ y = w) -> <.x, y>. = <.z, w>.)
14	13	eqeq2d 1486	. . . . 5 |- ((x = z /\ y = w) -> (v = <.x, y>. <-> v = <.z, w>.))
15		cbvopab.5	. . . . 5 |- ((x = z /\ y = w) -> (ph <-> ps))
16	14, 15	anbi12d 628	. . . 4 |- ((x = z /\ y = w) -> ((v = <.x, y>. /\ ph) <-> (v = <.z, w>. /\ ps)))
17	3, 6, 9, 12, 16	cbvex2 1317	. . 3 |- (E.xE.y(v = <.x, y>. /\ ph) <-> E.zE.w(v = <.z, w>. /\ ps))
18	17	abbii 1575	. 2 |- {v | E.xE.y(v = <.x, y>. /\ ph)} = {v | E.zE.w(v = <.z, w>. /\ ps)}
19		df-opab 2667	. 2 |- {<.x, y>. | ph} = {v | E.xE.y(v = <.x, y>. /\ ph)}
20		df-opab 2667	. 2 |- {<.z, w>. | ps} = {v | E.zE.w(v = <.z, w>. /\ ps)}
21	18, 19, 20	3eqtr4 1505	1 |- {<.x, y>. | ph} = {<.z, w>. | ps}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956  E.wex 980  {cab 1463  <.cop 2411  {copab 2666

This theorem is referenced by:  cbvopabv 2673  fvopab4gf 3781  fvopabgf 3787  fvopabnf 3788  cnvtr 10638

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-opab 2667

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