Theorem cbvopab2v 2672

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 2484	. . . . . . 7 |- (y = z -> <.x, y>. = <.x, z>.)
2	1	eqeq2d 1483	. . . . . 6 |- (y = z -> (w = <.x, y>. <-> w = <.x, z>.))
3		cbvopab2v.1	. . . . . 6 |- (y = z -> (ph <-> ps))
4	2, 3	anbi12d 627	. . . . 5 |- (y = z -> ((w = <.x, y>. /\ ph) <-> (w = <.x, z>. /\ ps)))
5	4	cbvexv 1313	. . . 4 |- (E.y(w = <.x, y>. /\ ph) <-> E.z(w = <.x, z>. /\ ps))
6	5	exbii 1049	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.xE.z(w = <.x, z>. /\ ps))
7	6	abbii 1572	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.xE.z(w = <.x, z>. /\ ps)}
8		df-opab 2662	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
9		df-opab 2662	. 2 |- {<.x, z>. | ps} = {w | E.xE.z(w = <.x, z>. /\ ps)}
10	7, 8, 9	3eqtr4 1502	1 |- {<.x, y>. | ph} = {<.x, z>. | ps}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954  E.wex 978  {cab 1461  <.cop 2407  {copab 2661

This theorem is referenced by:  cbvoprab3v 3991  ac6 4735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662

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