Theorem dvelimALT 2026

Description: Version of dvelim 2033 that doesn't use ax-10 1516. Because it has different distinct variable constraints than dvelim 2033 and is used in important proofs, it would be better if it had a name which does not end in ALT (ideally more close to set.mm naming). (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dvelimALT.1	|- ( ph -> A. x ph )
dvelimALT.2	|- ( z = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
dvelimALT	|- ( -. A. x x = y -> ( ps -> A. x ps ) )

Distinct variable groups:   ps, z   x, z   y, z

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

Proof of Theorem dvelimALT

Step	Hyp	Ref	Expression
1		nfv 1539	. . . 4 |- F/ z -. A. x x = y
2		ax12or 1519	. . . . . . . . 9 |- ( A. x x = z \/ ( A. x x = y \/ A. x ( z = y -> A. x z = y ) ) )
3		orcom 729	. . . . . . . . . 10 |- ( ( A. x x = y \/ A. x ( z = y -> A. x z = y ) ) <-> ( A. x ( z = y -> A. x z = y ) \/ A. x x = y ) )
4	3	orbi2i 763	. . . . . . . . 9 |- ( ( A. x x = z \/ ( A. x x = y \/ A. x ( z = y -> A. x z = y ) ) ) <-> ( A. x x = z \/ ( A. x ( z = y -> A. x z = y ) \/ A. x x = y ) ) )
5	2, 4	mpbi 145	. . . . . . . 8 |- ( A. x x = z \/ ( A. x ( z = y -> A. x z = y ) \/ A. x x = y ) )
6		orass 768	. . . . . . . 8 |- ( ( ( A. x x = z \/ A. x ( z = y -> A. x z = y ) ) \/ A. x x = y ) <-> ( A. x x = z \/ ( A. x ( z = y -> A. x z = y ) \/ A. x x = y ) ) )
7	5, 6	mpbir 146	. . . . . . 7 |- ( ( A. x x = z \/ A. x ( z = y -> A. x z = y ) ) \/ A. x x = y )
8		nfa1 1552	. . . . . . . . . . 11 |- F/ x A. x x = z
9		ax16ALT 1870	. . . . . . . . . . 11 |- ( A. x x = z -> ( z = y -> A. x z = y ) )
10	8, 9	nfd 1534	. . . . . . . . . 10 |- ( A. x x = z -> F/ x z = y )
11		dvelimALT.1	. . . . . . . . . . . 12 |- ( ph -> A. x ph )
12	11	nfi 1473	. . . . . . . . . . 11 |- F/ x ph
13	12	a1i 9	. . . . . . . . . 10 |- ( A. x x = z -> F/ x ph )
14	10, 13	nfimd 1596	. . . . . . . . 9 |- ( A. x x = z -> F/ x ( z = y -> ph ) )
15		df-nf 1472	. . . . . . . . . 10 |- ( F/ x z = y <-> A. x ( z = y -> A. x z = y ) )
16		id 19	. . . . . . . . . . 11 |- ( F/ x z = y -> F/ x z = y )
17	12	a1i 9	. . . . . . . . . . 11 |- ( F/ x z = y -> F/ x ph )
18	16, 17	nfimd 1596	. . . . . . . . . 10 |- ( F/ x z = y -> F/ x ( z = y -> ph ) )
19	15, 18	sylbir 135	. . . . . . . . 9 |- ( A. x ( z = y -> A. x z = y ) -> F/ x ( z = y -> ph ) )
20	14, 19	jaoi 717	. . . . . . . 8 |- ( ( A. x x = z \/ A. x ( z = y -> A. x z = y ) ) -> F/ x ( z = y -> ph ) )
21	20	orim1i 761	. . . . . . 7 |- ( ( ( A. x x = z \/ A. x ( z = y -> A. x z = y ) ) \/ A. x x = y ) -> ( F/ x ( z = y -> ph ) \/ A. x x = y ) )
22	7, 21	ax-mp 5	. . . . . 6 |- ( F/ x ( z = y -> ph ) \/ A. x x = y )
23		orcom 729	. . . . . 6 |- ( ( F/ x ( z = y -> ph ) \/ A. x x = y ) <-> ( A. x x = y \/ F/ x ( z = y -> ph ) ) )
24	22, 23	mpbi 145	. . . . 5 |- ( A. x x = y \/ F/ x ( z = y -> ph ) )
25	24	ori 724	. . . 4 |- ( -. A. x x = y -> F/ x ( z = y -> ph ) )
26	1, 25	nfald 1771	. . 3 |- ( -. A. x x = y -> F/ x A. z ( z = y -> ph ) )
27		ax-17 1537	. . . . 5 |- ( ps -> A. z ps )
28		dvelimALT.2	. . . . 5 |- ( z = y -> ( ph <-> ps ) )
29	27, 28	equsalh 1737	. . . 4 |- ( A. z ( z = y -> ph ) <-> ps )
30	29	nfbii 1484	. . 3 |- ( F/ x A. z ( z = y -> ph ) <-> F/ x ps )
31	26, 30	sylib 122	. 2 |- ( -. A. x x = y -> F/ x ps )
32	31	nfrd 1531	1 |- ( -. A. x x = y -> ( ps -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 709   A.wal 1362   F/wnf 1471

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-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  hbsb4  2028