Theorem dvelimor 2018

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 1509	. . . . . 6 |- ( A. x x = z \/ ( A. x x = y \/ A. z A. x ( z = y -> A. x z = y ) ) )
2		orcom 728	. . . . . . 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 762	. . . . . 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 145	. . . . 5 |- ( A. x x = z \/ ( A. z A. x ( z = y -> A. x z = y ) \/ A. x x = y ) )
5		orass 767	. . . . 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 146	. . . 4 |- ( ( A. x x = z \/ A. z A. x ( z = y -> A. x z = y ) ) \/ A. x x = y )
7		nfae 1719	. . . . . . 7 |- F/ z A. x x = z
8		a16nf 1866	. . . . . . 7 |- ( A. x x = z -> F/ x ( z = y -> ph ) )
9	7, 8	alrimi 1522	. . . . . 6 |- ( A. x x = z -> A. z F/ x ( z = y -> ph ) )
10		df-nf 1461	. . . . . . . 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 1585	. . . . . . . 8 |- ( F/ x z = y -> F/ x ( z = y -> ph ) )
15	10, 14	sylbir 135	. . . . . . 7 |- ( A. x ( z = y -> A. x z = y ) -> F/ x ( z = y -> ph ) )
16	15	alimi 1455	. . . . . 6 |- ( A. z A. x ( z = y -> A. x z = y ) -> A. z F/ x ( z = y -> ph ) )
17	9, 16	jaoi 716	. . . . 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 760	. . . 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 5	. . 3 |- ( A. z F/ x ( z = y -> ph ) \/ A. x x = y )
20		orcom 728	. . 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 145	. 2 |- ( A. x x = y \/ A. z F/ x ( z = y -> ph ) )
22		nfalt 1578	. . . 4 |- ( A. z F/ x ( z = y -> ph ) -> F/ x A. z ( z = y -> ph ) )
23		ax-17 1526	. . . . . 6 |- ( ps -> A. z ps )
24		dvelimor.2	. . . . . 6 |- ( z = y -> ( ph <-> ps ) )
25	23, 24	equsalh 1726	. . . . 5 |- ( A. z ( z = y -> ph ) <-> ps )
26	25	nfbii 1473	. . . 4 |- ( F/ x A. z ( z = y -> ph ) <-> F/ x ps )
27	22, 26	sylib 122	. . 3 |- ( A. z F/ x ( z = y -> ph ) -> F/ x ps )
28	27	orim2i 761	. 2 |- ( ( A. x x = y \/ A. z F/ x ( z = y -> ph ) ) -> ( A. x x = y \/ F/ x ps ) )
29	21, 28	ax-mp 5	1 |- ( A. x x = y \/ F/ x ps )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 708   A.wal 1351   F/wnf 1460

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  nfsb4or  2021  rgen2a  2531