Theorem cbvoprab2 6383

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 1769	. . . . . . 7 |- F/ w v = <. <. x , y >. , z >.
2		cbvoprab2.1	. . . . . . 7 |- F/ w ph
3	1, 2	nfan 2031	. . . . . 6 |- F/ w ( v = <. <. x , y >. , z >. /\ ph )
4	3	nfex 2050	. . . . 5 |- F/ w E. z ( v = <. <. x , y >. , z >. /\ ph )
5		nfv 1769	. . . . . . 7 |- F/ y v = <. <. x , w >. , z >.
6		cbvoprab2.2	. . . . . . 7 |- F/ y ps
7	5, 6	nfan 2031	. . . . . 6 |- F/ y ( v = <. <. x , w >. , z >. /\ ps )
8	7	nfex 2050	. . . . 5 |- F/ y E. z ( v = <. <. x , w >. , z >. /\ ps )
9		opeq2 4159	. . . . . . . . 9 |- ( y = w -> <. x , y >. = <. x , w >. )
10	9	opeq1d 4164	. . . . . . . 8 |- ( y = w -> <. <. x , y >. , z >. = <. <. x , w >. , z >. )
11	10	eqeq2d 2481	. . . . . . 7 |- ( y = w -> ( v = <. <. x , y >. , z >. <-> v = <. <. x , w >. , z >. ) )
12		cbvoprab2.3	. . . . . . 7 |- ( y = w -> ( ph <-> ps ) )
13	11, 12	anbi12d 725	. . . . . 6 |- ( y = w -> ( ( v = <. <. x , y >. , z >. /\ ph ) <-> ( v = <. <. x , w >. , z >. /\ ps ) ) )
14	13	exbidv 1776	. . . . 5 |- ( y = w -> ( E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. z ( v = <. <. x , w >. , z >. /\ ps ) ) )
15	4, 8, 14	cbvex 2128	. . . 4 |- ( E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) )
16	15	exbii 1726	. . 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 2587	. 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 6312	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
19		df-oprab 6312	. 2 |- { <. <. x , w >. , z >. | ps } = { v | E. x E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) }
20	17, 18, 19	3eqtr4i 2503	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , w >. , z >. | ps }

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   F/wnf 1675   {cab 2457   <.cop 3965   {coprab 6309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-oprab 6312

This theorem is referenced by: (None)