Theorem bdsepnft 9342

Description: Closed form of bdsepnf 9343. Version of ax-bdsep 9339 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 9340 when sufficient. (Contributed by BJ, 19-Oct-2019.)

Hypothesis

Ref	Expression
bdsepnft.1	|- BOUNDED ph

Assertion

Ref	Expression
bdsepnft	|- (A.x F/ bph -> E. bA.x(x e. b <-> (x e. a /\ ph)))

Distinct variable group:   a, b,x

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

Proof of Theorem bdsepnft

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdsepnft.1	. . 3 |- BOUNDED ph
2	1	bdsep2 9341	. 2 |- E.yA.x(x e. y <-> (x e. a /\ ph))
3		nfnf1 1433	. . . 4 |- F/ b F/ bph
4	3	nfal 1465	. . 3 |- F/ bA.x F/ bph
5		nfa1 1431	. . . 4 |- F/xA.x F/ bph
6		nfvd 1419	. . . . 5 |- (A.x F/ bph -> F/ b x e. y)
7		nfv 1418	. . . . . . 7 |- F/ b x e. a
8	7	a1i 9	. . . . . 6 |- (A.x F/ bph -> F/ b x e. a)
9		sp 1398	. . . . . 6 |- (A.x F/ bph -> F/ bph)
10	8, 9	nfand 1457	. . . . 5 |- (A.x F/ bph -> F/ b(x e. a /\ ph))
11	6, 10	nfbid 1477	. . . 4 |- (A.x F/ bph -> F/ b(x e. y <-> (x e. a /\ ph)))
12	5, 11	nfald 1640	. . 3 |- (A.x F/ bph -> F/ bA.x(x e. y <-> (x e. a /\ ph)))
13		nfv 1418	. . . . . 6 |- F/x y = b
14	5, 13	nfan 1454	. . . . 5 |- F/x(A.x F/ bph /\ y = b)
15		elequ2 1598	. . . . . . 7 |- (y = b -> (x e. y <-> x e. b))
16	15	adantl 262	. . . . . 6 |- ((A.x F/ bph /\ y = b) -> (x e. y <-> x e. b))
17	16	bibi1d 222	. . . . 5 |- ((A.x F/ bph /\ y = b) -> ((x e. y <-> (x e. a /\ ph)) <-> (x e. b <-> (x e. a /\ ph))))
18	14, 17	albid 1503	. . . 4 |- ((A.x F/ bph /\ y = b) -> (A.x(x e. y <-> (x e. a /\ ph)) <-> A.x(x e. b <-> (x e. a /\ ph))))
19	18	ex 108	. . 3 |- (A.x F/ bph -> (y = b -> (A.x(x e. y <-> (x e. a /\ ph)) <-> A.x(x e. b <-> (x e. a /\ ph)))))
20	4, 12, 19	cbvexd 1799	. 2 |- (A.x F/ bph -> (E.yA.x(x e. y <-> (x e. a /\ ph)) <-> E. bA.x(x e. b <-> (x e. a /\ ph))))
21	2, 20	mpbii 136	1 |- (A.x F/ bph -> E. bA.x(x e. b <-> (x e. a /\ ph)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   F/wnf 1346  E.wex 1378  BOUNDED wbd 9267

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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bdsep 9339

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033

This theorem is referenced by:  bdsepnf  9343