Theorem strcollnft 9414

Description: Closed form of strcollnf 9415. Version of ax-strcoll 9412 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)

Assertion

Ref	Expression
strcollnft	|- (A.xA.y F/ bph -> (A.x e. a E.yph -> E. bA.y(y e. b <-> E.x e. a ph)))

Distinct variable group:   a, b,x,y

Allowed substitution hints:   ph(x,y, a, b)

Proof of Theorem strcollnft

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		strcoll2 9413	. 2 |- (A.x e. a E.yph -> E.zA.y(y e. z <-> E.x e. a ph))
2		nfnf1 1433	. . . . 5 |- F/ b F/ bph
3	2	nfal 1465	. . . 4 |- F/ bA.y F/ bph
4	3	nfal 1465	. . 3 |- F/ bA.xA.y F/ bph
5		nfa2 1468	. . . 4 |- F/yA.xA.y F/ bph
6		nfvd 1419	. . . . 5 |- (A.xA.y F/ bph -> F/ b y e. z)
7		nfa1 1431	. . . . . . . 8 |- F/xA.x F/ bph
8		nfcvd 2176	. . . . . . . 8 |- (A.x F/ bph -> F/_ b a)
9		sp 1398	. . . . . . . 8 |- (A.x F/ bph -> F/ bph)
10	7, 8, 9	nfrexdxy 2351	. . . . . . 7 |- (A.x F/ bph -> F/ bE.x e. a ph)
11	10	sps 1427	. . . . . 6 |- (A.yA.x F/ bph -> F/ bE.x e. a ph)
12	11	alcoms 1362	. . . . 5 |- (A.xA.y F/ bph -> F/ bE.x e. a ph)
13	6, 12	nfbid 1477	. . . 4 |- (A.xA.y F/ bph -> F/ b(y e. z <-> E.x e. a ph))
14	5, 13	nfald 1640	. . 3 |- (A.xA.y F/ bph -> F/ bA.y(y e. z <-> E.x e. a ph))
15		nfv 1418	. . . . . 6 |- F/y z = b
16	5, 15	nfan 1454	. . . . 5 |- F/y(A.xA.y F/ bph /\ z = b)
17		elequ2 1598	. . . . . . 7 |- (z = b -> (y e. z <-> y e. b))
18	17	adantl 262	. . . . . 6 |- ((A.xA.y F/ bph /\ z = b) -> (y e. z <-> y e. b))
19	18	bibi1d 222	. . . . 5 |- ((A.xA.y F/ bph /\ z = b) -> ((y e. z <-> E.x e. a ph) <-> (y e. b <-> E.x e. a ph)))
20	16, 19	albid 1503	. . . 4 |- ((A.xA.y F/ bph /\ z = b) -> (A.y(y e. z <-> E.x e. a ph) <-> A.y(y e. b <-> E.x e. a ph)))
21	20	ex 108	. . 3 |- (A.xA.y F/ bph -> (z = b -> (A.y(y e. z <-> E.x e. a ph) <-> A.y(y e. b <-> E.x e. a ph))))
22	4, 14, 21	cbvexd 1799	. 2 |- (A.xA.y F/ bph -> (E.zA.y(y e. z <-> E.x e. a ph) <-> E. bA.y(y e. b <-> E.x e. a ph)))
23	1, 22	syl5ib 143	1 |- (A.xA.y F/ bph -> (A.x e. a E.yph -> E. bA.y(y e. b <-> E.x e. a ph)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   F/wnf 1346  E.wex 1378  A.wral 2300  E.wrex 2301

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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-strcoll 9412

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306

This theorem is referenced by:  strcollnf  9415