Theorem cbvopab2v 2109

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 group(s):   x,y,z   ph,z   ps,y

Proof of Theorem cbvopab2v

Step	Hyp	Ref	Expression
1		opeq2 1877	. . . . . . 7 |- (y = z -> <.x, y>. = <.x, z>.)
2	1	cleq2d 1112	. . . . . 6 |- (y = z -> (w = <.x, y>. <-> w = <.x, z>.))
3		cbvopab2v.1	. . . . . 6 |- (y = z -> (ph <-> ps))
4	2, 3	anbi12d 476	. . . . 5 |- (y = z -> ((w = <.x, y>. /\ ph) <-> (w = <.x, z>. /\ ps)))
5	4	cbvexv 973	. . . 4 |- (E.y(w = <.x, y>. /\ ph) <-> E.z(w = <.x, z>. /\ ps))
6	5	biex 733	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.xE.z(w = <.x, z>. /\ ps))
7	6	biabi 1181	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.xE.z(w = <.x, z>. /\ ps)}
8		df-opab 2098	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
9		df-opab 2098	. 2 |- {<.x, z>. | ps} = {w | E.xE.z(w = <.x, z>. /\ ps)}
10	7, 8, 9	3eqtr4 1126	1 |- {<.x, y>. | ph} = {<.x, z>. | ps}

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = weq 797  {cab 1090   = wceq 1091  <.cop 1810  {copab 2055

This theorem is referenced by:  cbvoprab3v 3030  ac6 3576

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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