Theorem cbvoprab3v 3991

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution.

Hypothesis

Ref	Expression
cbvoprab3v.1	|- (z = w -> (ph <-> ps))

Assertion

Ref	Expression
cbvoprab3v	|- {<.<.x, y>., z>. | ph} = {<.<.x, y>., w>. | ps}

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

Proof of Theorem cbvoprab3v

Step	Hyp	Ref	Expression
1		cbvoprab3v.1	. . . . 5 |- (z = w -> (ph <-> ps))
2	1	anbi2d 615	. . . 4 |- (z = w -> ((v = <.x, y>. /\ ph) <-> (v = <.x, y>. /\ ps)))
3	2	2exbidv 1279	. . 3 |- (z = w -> (E.xE.y(v = <.x, y>. /\ ph) <-> E.xE.y(v = <.x, y>. /\ ps)))
4	3	cbvopab2v 2672	. 2 |- {<.v, z>. | E.xE.y(v = <.x, y>. /\ ph)} = {<.v, w>. | E.xE.y(v = <.x, y>. /\ ps)}
5		dfoprab2 3982	. 2 |- {<.<.x, y>., z>. | ph} = {<.v, z>. | E.xE.y(v = <.x, y>. /\ ph)}
6		dfoprab2 3982	. 2 |- {<.<.x, y>., w>. | ps} = {<.v, w>. | E.xE.y(v = <.x, y>. /\ ps)}
7	4, 5, 6	3eqtr4 1502	1 |- {<.<.x, y>., z>. | ph} = {<.<.x, y>., w>. | ps}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954  E.wex 978  <.cop 2407  {copab 2661  {copab2 3955

This theorem is referenced by:  acdc3 7437  acdc2 7440  acdc5 7443  acdc 7445  ruclem12 7472

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-11 965  ax-12 966  ax-13 967  ax-14 968  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  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662  df-oprab 3957

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