Theorem dvelimALT 1927

Description: Version of dvelim 1934 that doesn't use ax-10 1436. Because it has different distinct variable constraints than dvelim 1934 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 1461	. . . 4 |- F/ z -. A. x x = y
2		ax-i12 1438	. . . . . . . . 9 |- ( A. x x = z \/ ( A. x x = y \/ A. x ( z = y -> A. x z = y ) ) )
3		orcom 679	. . . . . . . . . 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 711	. . . . . . . . 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 143	. . . . . . . 8 |- ( A. x x = z \/ ( A. x ( z = y -> A. x z = y ) \/ A. x x = y ) )
6		orass 716	. . . . . . . 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 144	. . . . . . 7 |- ( ( A. x x = z \/ A. x ( z = y -> A. x z = y ) ) \/ A. x x = y )
8		nfa1 1474	. . . . . . . . . . 11 |- F/ x A. x x = z
9		ax16ALT 1780	. . . . . . . . . . 11 |- ( A. x x = z -> ( z = y -> A. x z = y ) )
10	8, 9	nfd 1456	. . . . . . . . . 10 |- ( A. x x = z -> F/ x z = y )
11		dvelimALT.1	. . . . . . . . . . . 12 |- ( ph -> A. x ph )
12	11	nfi 1391	. . . . . . . . . . 11 |- F/ x ph
13	12	a1i 9	. . . . . . . . . 10 |- ( A. x x = z -> F/ x ph )
14	10, 13	nfimd 1517	. . . . . . . . 9 |- ( A. x x = z -> F/ x ( z = y -> ph ) )
15		df-nf 1390	. . . . . . . . . 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 1517	. . . . . . . . . 10 |- ( F/ x z = y -> F/ x ( z = y -> ph ) )
19	15, 18	sylbir 133	. . . . . . . . 9 |- ( A. x ( z = y -> A. x z = y ) -> F/ x ( z = y -> ph ) )
20	14, 19	jaoi 668	. . . . . . . 8 |- ( ( A. x x = z \/ A. x ( z = y -> A. x z = y ) ) -> F/ x ( z = y -> ph ) )
21	20	orim1i 709	. . . . . . 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 7	. . . . . 6 |- ( F/ x ( z = y -> ph ) \/ A. x x = y )
23		orcom 679	. . . . . 6 |- ( ( F/ x ( z = y -> ph ) \/ A. x x = y ) <-> ( A. x x = y \/ F/ x ( z = y -> ph ) ) )
24	22, 23	mpbi 143	. . . . 5 |- ( A. x x = y \/ F/ x ( z = y -> ph ) )
25	24	ori 674	. . . 4 |- ( -. A. x x = y -> F/ x ( z = y -> ph ) )
26	1, 25	nfald 1683	. . 3 |- ( -. A. x x = y -> F/ x A. z ( z = y -> ph ) )
27		ax-17 1459	. . . . 5 |- ( ps -> A. z ps )
28		dvelimALT.2	. . . . 5 |- ( z = y -> ( ph <-> ps ) )
29	27, 28	equsalh 1654	. . . 4 |- ( A. z ( z = y -> ph ) <-> ps )
30	29	nfbii 1402	. . 3 |- ( F/ x A. z ( z = y -> ph ) <-> F/ x ps )
31	26, 30	sylib 120	. 2 |- ( -. A. x x = y -> F/ x ps )
32	31	nfrd 1453	1 |- ( -. A. x x = y -> ( ps -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661   A.wal 1282   F/wnf 1389

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-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  hbsb4  1929