Theorem dvelimor 1891

Description: Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula ph (containing z) and a distinct variable constraint between x and z. The theorem makes it possible to replace the distinct variable constraint with the disjunct A.xx = y (ps is just a version of ph with y substituted for z). (Contributed by Jim Kingdon, 11-May-2018.)

Hypotheses

Ref	Expression
dvelimor.1	|- F/xph
dvelimor.2	|- (z = y -> (ph <-> ps))

Assertion

Ref	Expression
dvelimor	|- (A.x x = y \/ F/xps)

Distinct variable groups:   ps,z   x,z

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

Proof of Theorem dvelimor

Step	Hyp	Ref	Expression
1		ax-bnd 1396	. . . . . 6 |- (A.x x = z \/ (A.x x = y \/ A.zA.x(z = y -> A.x z = y)))
2		orcom 646	. . . . . . 7 |- ((A.x x = y \/ A.zA.x(z = y -> A.x z = y)) <-> (A.zA.x(z = y -> A.x z = y) \/ A.x x = y))
3	2	orbi2i 678	. . . . . 6 |- ((A.x x = z \/ (A.x x = y \/ A.zA.x(z = y -> A.x z = y))) <-> (A.x x = z \/ (A.zA.x(z = y -> A.x z = y) \/ A.x x = y)))
4	1, 3	mpbi 133	. . . . 5 |- (A.x x = z \/ (A.zA.x(z = y -> A.x z = y) \/ A.x x = y))
5		orass 683	. . . . 5 |- (((A.x x = z \/ A.zA.x(z = y -> A.x z = y)) \/ A.x x = y) <-> (A.x x = z \/ (A.zA.x(z = y -> A.x z = y) \/ A.x x = y)))
6	4, 5	mpbir 134	. . . 4 |- ((A.x x = z \/ A.zA.x(z = y -> A.x z = y)) \/ A.x x = y)
7		nfae 1604	. . . . . . 7 |- F/zA.x x = z
8		a16nf 1743	. . . . . . 7 |- (A.x x = z -> F/x(z = y -> ph))
9	7, 8	alrimi 1412	. . . . . 6 |- (A.x x = z -> A.z F/x(z = y -> ph))
10		df-nf 1347	. . . . . . . 8 |- ( F/x z = y <-> A.x(z = y -> A.x z = y))
11		id 19	. . . . . . . . 9 |- ( F/x z = y -> F/x z = y)
12		dvelimor.1	. . . . . . . . . 10 |- F/xph
13	12	a1i 9	. . . . . . . . 9 |- ( F/x z = y -> F/xph)
14	11, 13	nfimd 1474	. . . . . . . 8 |- ( F/x z = y -> F/x(z = y -> ph))
15	10, 14	sylbir 125	. . . . . . 7 |- (A.x(z = y -> A.x z = y) -> F/x(z = y -> ph))
16	15	alimi 1341	. . . . . 6 |- (A.zA.x(z = y -> A.x z = y) -> A.z F/x(z = y -> ph))
17	9, 16	jaoi 635	. . . . 5 |- ((A.x x = z \/ A.zA.x(z = y -> A.x z = y)) -> A.z F/x(z = y -> ph))
18	17	orim1i 676	. . . 4 |- (((A.x x = z \/ A.zA.x(z = y -> A.x z = y)) \/ A.x x = y) -> (A.z F/x(z = y -> ph) \/ A.x x = y))
19	6, 18	ax-mp 7	. . 3 |- (A.z F/x(z = y -> ph) \/ A.x x = y)
20		orcom 646	. . 3 |- ((A.z F/x(z = y -> ph) \/ A.x x = y) <-> (A.x x = y \/ A.z F/x(z = y -> ph)))
21	19, 20	mpbi 133	. 2 |- (A.x x = y \/ A.z F/x(z = y -> ph))
22		nfalt 1467	. . . 4 |- (A.z F/x(z = y -> ph) -> F/xA.z(z = y -> ph))
23		ax-17 1416	. . . . . 6 |- (ps -> A.zps)
24		dvelimor.2	. . . . . 6 |- (z = y -> (ph <-> ps))
25	23, 24	equsalh 1611	. . . . 5 |- (A.z(z = y -> ph) <-> ps)
26	25	nfbii 1359	. . . 4 |- ( F/xA.z(z = y -> ph) <-> F/xps)
27	22, 26	sylib 127	. . 3 |- (A.z F/x(z = y -> ph) -> F/xps)
28	27	orim2i 677	. 2 |- ((A.x x = y \/ A.z F/x(z = y -> ph)) -> (A.x x = y \/ F/xps))
29	21, 28	ax-mp 7	1 |- (A.x x = y \/ F/xps)

Syntax hints:   -> wi 4   <-> wb 98   \/ wo 628  A.wal 1240   F/wnf 1346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  nfsb4or  1896  rgen2a  2369