Theorem dvelimor 1942

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. x x = 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/ x ph
dvelimor.2	|- ( z = y -> ( ph <-> ps ) )

Assertion

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

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-bndl 1444	. . . . . 6 |- ( A. x x = z \/ ( A. x x = y \/ A. z A. x ( z = y -> A. x z = y ) ) )
2		orcom 682	. . . . . . 7 |- ( ( A. x x = y \/ A. z A. x ( z = y -> A. x z = y ) ) <-> ( A. z A. x ( z = y -> A. x z = y ) \/ A. x x = y ) )
3	2	orbi2i 714	. . . . . 6 |- ( ( A. x x = z \/ ( A. x x = y \/ A. z A. x ( z = y -> A. x z = y ) ) ) <-> ( A. x x = z \/ ( A. z A. x ( z = y -> A. x z = y ) \/ A. x x = y ) ) )
4	1, 3	mpbi 143	. . . . 5 |- ( A. x x = z \/ ( A. z A. x ( z = y -> A. x z = y ) \/ A. x x = y ) )
5		orass 719	. . . . 5 |- ( ( ( A. x x = z \/ A. z A. x ( z = y -> A. x z = y ) ) \/ A. x x = y ) <-> ( A. x x = z \/ ( A. z A. x ( z = y -> A. x z = y ) \/ A. x x = y ) ) )
6	4, 5	mpbir 144	. . . 4 |- ( ( A. x x = z \/ A. z A. x ( z = y -> A. x z = y ) ) \/ A. x x = y )
7		nfae 1654	. . . . . . 7 |- F/ z A. x x = z
8		a16nf 1794	. . . . . . 7 |- ( A. x x = z -> F/ x ( z = y -> ph ) )
9	7, 8	alrimi 1460	. . . . . 6 |- ( A. x x = z -> A. z F/ x ( z = y -> ph ) )
10		df-nf 1395	. . . . . . . 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/ x ph
13	12	a1i 9	. . . . . . . . 9 |- ( F/ x z = y -> F/ x ph )
14	11, 13	nfimd 1522	. . . . . . . 8 |- ( F/ x z = y -> F/ x ( z = y -> ph ) )
15	10, 14	sylbir 133	. . . . . . 7 |- ( A. x ( z = y -> A. x z = y ) -> F/ x ( z = y -> ph ) )
16	15	alimi 1389	. . . . . 6 |- ( A. z A. x ( z = y -> A. x z = y ) -> A. z F/ x ( z = y -> ph ) )
17	9, 16	jaoi 671	. . . . 5 |- ( ( A. x x = z \/ A. z A. x ( z = y -> A. x z = y ) ) -> A. z F/ x ( z = y -> ph ) )
18	17	orim1i 712	. . . 4 |- ( ( ( A. x x = z \/ A. z A. 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 682	. . 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 143	. 2 |- ( A. x x = y \/ A. z F/ x ( z = y -> ph ) )
22		nfalt 1515	. . . 4 |- ( A. z F/ x ( z = y -> ph ) -> F/ x A. z ( z = y -> ph ) )
23		ax-17 1464	. . . . . 6 |- ( ps -> A. z ps )
24		dvelimor.2	. . . . . 6 |- ( z = y -> ( ph <-> ps ) )
25	23, 24	equsalh 1661	. . . . 5 |- ( A. z ( z = y -> ph ) <-> ps )
26	25	nfbii 1407	. . . 4 |- ( F/ x A. z ( z = y -> ph ) <-> F/ x ps )
27	22, 26	sylib 120	. . 3 |- ( A. z F/ x ( z = y -> ph ) -> F/ x ps )
28	27	orim2i 713	. 2 |- ( ( A. x x = y \/ A. z F/ x ( z = y -> ph ) ) -> ( A. x x = y \/ F/ x ps ) )
29	21, 28	ax-mp 7	1 |- ( A. x x = y \/ F/ x ps )

Syntax hints:   -> wi 4   <-> wb 103   \/ wo 664   A.wal 1287   F/wnf 1394

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693

This theorem is referenced by:  nfsb4or  1947  rgen2a  2429