Theorem cbvoprab2 5569

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/wph
cbvoprab2.2	|- F/yps
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 1619	. . . . . . 7 |- F/w v = <.<.x, y>., z>.
2		cbvoprab2.1	. . . . . . 7 |- F/wph
3	1, 2	nfan 1824	. . . . . 6 |- F/w(v = <.<.x, y>., z>. /\ ph)
4	3	nfex 1843	. . . . 5 |- F/wE.z(v = <.<.x, y>., z>. /\ ph)
5		nfv 1619	. . . . . . 7 |- F/y v = <.<.x, w>., z>.
6		cbvoprab2.2	. . . . . . 7 |- F/yps
7	5, 6	nfan 1824	. . . . . 6 |- F/y(v = <.<.x, w>., z>. /\ ps)
8	7	nfex 1843	. . . . 5 |- F/yE.z(v = <.<.x, w>., z>. /\ ps)
9		opeq2 4580	. . . . . . . . 9 |- (y = w -> <.x, y>. = <.x, w>.)
10	9	opeq1d 4585	. . . . . . . 8 |- (y = w -> <.<.x, y>., z>. = <.<.x, w>., z>.)
11	10	eqeq2d 2364	. . . . . . 7 |- (y = w -> (v = <.<.x, y>., z>. <-> v = <.<.x, w>., z>.))
12		cbvoprab2.3	. . . . . . 7 |- (y = w -> (ph <-> ps))
13	11, 12	anbi12d 691	. . . . . 6 |- (y = w -> ((v = <.<.x, y>., z>. /\ ph) <-> (v = <.<.x, w>., z>. /\ ps)))
14	13	exbidv 1626	. . . . 5 |- (y = w -> (E.z(v = <.<.x, y>., z>. /\ ph) <-> E.z(v = <.<.x, w>., z>. /\ ps)))
15	4, 8, 14	cbvex 1985	. . . 4 |- (E.yE.z(v = <.<.x, y>., z>. /\ ph) <-> E.wE.z(v = <.<.x, w>., z>. /\ ps))
16	15	exbii 1582	. . 3 |- (E.xE.yE.z(v = <.<.x, y>., z>. /\ ph) <-> E.xE.wE.z(v = <.<.x, w>., z>. /\ ps))
17	16	abbii 2466	. 2 |- {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)} = {v | E.xE.wE.z(v = <.<.x, w>., z>. /\ ps)}
18		df-oprab 5529	. 2 |- {<.<.x, y>., z>. | ph} = {v | E.xE.yE.z(v = <.<.x, y>., z>. /\ ph)}
19		df-oprab 5529	. 2 |- {<.<.x, w>., z>. | ps} = {v | E.xE.wE.z(v = <.<.x, w>., z>. /\ ps)}
20	17, 18, 19	3eqtr4i 2383	1 |- {<.<.x, y>., z>. | ph} = {<.<.x, w>., z>. | ps}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642  {cab 2339  <.cop 4562  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-oprab 5529

This theorem is referenced by: (None)