Theorem cbvopab2v 2677

Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution.

Hypothesis

Ref	Expression
cbvopab2v.1	|- (y = z -> (ph <-> ps))

Assertion

Ref	Expression
cbvopab2v	|- {<.x, y>. | ph} = {<.x, z>. | ps}

Distinct variable groups:   x,y,z   ph,z   ps,y

Proof of Theorem cbvopab2v

Step	Hyp	Ref	Expression
1		opeq2 2488	. . . . . . 7 |- (y = z -> <.x, y>. = <.x, z>.)
2	1	eqeq2d 1486	. . . . . 6 |- (y = z -> (w = <.x, y>. <-> w = <.x, z>.))
3		cbvopab2v.1	. . . . . 6 |- (y = z -> (ph <-> ps))
4	2, 3	anbi12d 628	. . . . 5 |- (y = z -> ((w = <.x, y>. /\ ph) <-> (w = <.x, z>. /\ ps)))
5	4	cbvexv 1315	. . . 4 |- (E.y(w = <.x, y>. /\ ph) <-> E.z(w = <.x, z>. /\ ps))
6	5	exbii 1051	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.xE.z(w = <.x, z>. /\ ps))
7	6	abbii 1575	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.xE.z(w = <.x, z>. /\ ps)}
8		df-opab 2667	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
9		df-opab 2667	. 2 |- {<.x, z>. | ps} = {w | E.xE.z(w = <.x, z>. /\ ps)}
10	7, 8, 9	3eqtr4 1505	1 |- {<.x, y>. | ph} = {<.x, z>. | ps}

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

This theorem is referenced by:  cbvoprab3v 4000  ac6 4755

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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