Theorem cbvoprab2 6018

Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
cbvoprab2.1	|- F/ w ph
cbvoprab2.2	|- F/ y ps
cbvoprab2.3	|- ( y = w -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvoprab2	|- { <. <. x , y >. , z >. | ph } = { <. <. x , w >. , z >. | ps }

Distinct variable group:   x, w, y, z

Allowed substitution hints:   ph( x, y, z, w)   ps( x, y, z, w)

Proof of Theorem cbvoprab2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1551	. . . . . . 7 |- F/ w v = <. <. x , y >. , z >.
2		cbvoprab2.1	. . . . . . 7 |- F/ w ph
3	1, 2	nfan 1588	. . . . . 6 |- F/ w ( v = <. <. x , y >. , z >. /\ ph )
4	3	nfex 1660	. . . . 5 |- F/ w E. z ( v = <. <. x , y >. , z >. /\ ph )
5		nfv 1551	. . . . . . 7 |- F/ y v = <. <. x , w >. , z >.
6		cbvoprab2.2	. . . . . . 7 |- F/ y ps
7	5, 6	nfan 1588	. . . . . 6 |- F/ y ( v = <. <. x , w >. , z >. /\ ps )
8	7	nfex 1660	. . . . 5 |- F/ y E. z ( v = <. <. x , w >. , z >. /\ ps )
9		opeq2 3820	. . . . . . . . 9 |- ( y = w -> <. x , y >. = <. x , w >. )
10	9	opeq1d 3825	. . . . . . . 8 |- ( y = w -> <. <. x , y >. , z >. = <. <. x , w >. , z >. )
11	10	eqeq2d 2217	. . . . . . 7 |- ( y = w -> ( v = <. <. x , y >. , z >. <-> v = <. <. x , w >. , z >. ) )
12		cbvoprab2.3	. . . . . . 7 |- ( y = w -> ( ph <-> ps ) )
13	11, 12	anbi12d 473	. . . . . 6 |- ( y = w -> ( ( v = <. <. x , y >. , z >. /\ ph ) <-> ( v = <. <. x , w >. , z >. /\ ps ) ) )
14	13	exbidv 1848	. . . . 5 |- ( y = w -> ( E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. z ( v = <. <. x , w >. , z >. /\ ps ) ) )
15	4, 8, 14	cbvex 1779	. . . 4 |- ( E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) )
16	15	exbii 1628	. . 3 |- ( E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. x E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) )
17	16	abbii 2321	. 2 |- { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) } = { v | E. x E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) }
18		df-oprab 5948	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
19		df-oprab 5948	. 2 |- { <. <. x , w >. , z >. | ps } = { v | E. x E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) }
20	17, 18, 19	3eqtr4i 2236	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , w >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1483   E.wex 1515   {cab 2191   <.cop 3636   {coprab 5945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-oprab 5948

This theorem is referenced by: (None)