Theorem cbvopab1 2674

Description: Change first bound variable in an ordered-pair class 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:   x,y,z

Proof of Theorem cbvopab1

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . . 6 |- (w = <.x, y>. -> A.z w = <.x, y>.)
2		cbvopab1.1	. . . . . 6 |- (ph -> A.zph)
3	1, 2	hban 1009	. . . . 5 |- ((w = <.x, y>. /\ ph) -> A.z(w = <.x, y>. /\ ph))
4	3	hbex 1006	. . . 4 |- (E.y(w = <.x, y>. /\ ph) -> A.zE.y(w = <.x, y>. /\ ph))
5		ax-17 971	. . . . . 6 |- (w = <.z, y>. -> A.x w = <.z, y>.)
6		cbvopab1.2	. . . . . 6 |- (ps -> A.xps)
7	5, 6	hban 1009	. . . . 5 |- ((w = <.z, y>. /\ ps) -> A.x(w = <.z, y>. /\ ps))
8	7	hbex 1006	. . . 4 |- (E.y(w = <.z, y>. /\ ps) -> A.xE.y(w = <.z, y>. /\ ps))
9		opeq1 2487	. . . . . . 7 |- (x = z -> <.x, y>. = <.z, y>.)
10	9	eqeq2d 1486	. . . . . 6 |- (x = z -> (w = <.x, y>. <-> w = <.z, y>.))
11		cbvopab1.3	. . . . . 6 |- (x = z -> (ph <-> ps))
12	10, 11	anbi12d 628	. . . . 5 |- (x = z -> ((w = <.x, y>. /\ ph) <-> (w = <.z, y>. /\ ps)))
13	12	exbidv 1279	. . . 4 |- (x = z -> (E.y(w = <.x, y>. /\ ph) <-> E.y(w = <.z, y>. /\ ps)))
14	4, 8, 13	cbvex 1166	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.zE.y(w = <.z, y>. /\ ps))
15	14	abbii 1575	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.zE.y(w = <.z, y>. /\ ps)}
16		df-opab 2667	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
17		df-opab 2667	. 2 |- {<.z, y>. | ps} = {w | E.zE.y(w = <.z, y>. /\ ps)}
18	15, 16, 17	3eqtr4 1505	1 |- {<.x, y>. | ph} = {<.z, y>. | 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:  cbvopab1v 2676  seq1lem1 6309  cbvsum 6986

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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