Theorem cbvopab 2104

Description: Rule used to change bound variables in an ordered pair 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(s):   x,y,z,w

Proof of Theorem cbvopab

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- (v = <.x, y>. -> A.z v = <.x, y>.)
2		cbvopab.1	. . . . 5 |- (ph -> A.zph)
3	1, 2	hban 704	. . . 4 |- ((v = <.x, y>. /\ ph) -> A.z(v = <.x, y>. /\ ph))
4		ax-17 925	. . . . 5 |- (v = <.x, y>. -> A.w v = <.x, y>.)
5		cbvopab.2	. . . . 5 |- (ph -> A.wph)
6	4, 5	hban 704	. . . 4 |- ((v = <.x, y>. /\ ph) -> A.w(v = <.x, y>. /\ ph))
7		ax-17 925	. . . . 5 |- (v = <.z, w>. -> A.x v = <.z, w>.)
8		cbvopab.3	. . . . 5 |- (ps -> A.xps)
9	7, 8	hban 704	. . . 4 |- ((v = <.z, w>. /\ ps) -> A.x(v = <.z, w>. /\ ps))
10		ax-17 925	. . . . 5 |- (v = <.z, w>. -> A.y v = <.z, w>.)
11		cbvopab.4	. . . . 5 |- (ps -> A.yps)
12	10, 11	hban 704	. . . 4 |- ((v = <.z, w>. /\ ps) -> A.y(v = <.z, w>. /\ ps))
13		opeq12 1878	. . . . . 6 |- ((x = z /\ y = w) -> <.x, y>. = <.z, w>.)
14	13	cleq2d 1112	. . . . 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 476	. . . 4 |- ((x = z /\ y = w) -> ((v = <.x, y>. /\ ph) <-> (v = <.z, w>. /\ ps)))
17	3, 6, 9, 12, 16	cbvex2 975	. . 3 |- (E.xE.y(v = <.x, y>. /\ ph) <-> E.zE.w(v = <.z, w>. /\ ps))
18	17	biabi 1181	. 2 |- {v | E.xE.y(v = <.x, y>. /\ ph)} = {v | E.zE.w(v = <.z, w>. /\ ps)}
19		df-opab 2098	. 2 |- {<.x, y>. | ph} = {v | E.xE.y(v = <.x, y>. /\ ph)}
20		df-opab 2098	. 2 |- {<.z, w>. | ps} = {v | E.zE.w(v = <.z, w>. /\ ps)}
21	18, 19, 20	3eqtr4 1126	1 |- {<.x, y>. | ph} = {<.z, w>. | 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:  cbvopabv 2105  fvopabgf 2874  fvopabnf 2875

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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