Theorem bdsepnft 10007

Description: Closed form of bdsepnf 10008. Version of ax-bdsep 10004 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10005 when sufficient. (Contributed by BJ, 19-Oct-2019.)

Hypothesis

Ref	Expression
bdsepnft.1	|- BOUNDED ph

Assertion

Ref	Expression
bdsepnft	|- ( A. x F/ b ph -> E. b A. 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 10006	. 2 |- E. y A. x ( x e. y <-> ( x e. a /\ ph ) )
3		nfnf1 1436	. . . 4 |- F/ b F/ b ph
4	3	nfal 1468	. . 3 |- F/ b A. x F/ b ph
5		nfa1 1434	. . . 4 |- F/ x A. x F/ b ph
6		nfvd 1422	. . . . 5 |- ( A. x F/ b ph -> F/ b x e. y )
7		nfv 1421	. . . . . . 7 |- F/ b x e. a
8	7	a1i 9	. . . . . 6 |- ( A. x F/ b ph -> F/ b x e. a )
9		sp 1401	. . . . . 6 |- ( A. x F/ b ph -> F/ b ph )
10	8, 9	nfand 1460	. . . . 5 |- ( A. x F/ b ph -> F/ b ( x e. a /\ ph ) )
11	6, 10	nfbid 1480	. . . 4 |- ( A. x F/ b ph -> F/ b ( x e. y <-> ( x e. a /\ ph ) ) )
12	5, 11	nfald 1643	. . 3 |- ( A. x F/ b ph -> F/ b A. x ( x e. y <-> ( x e. a /\ ph ) ) )
13		nfv 1421	. . . . . 6 |- F/ x y = b
14	5, 13	nfan 1457	. . . . 5 |- F/ x ( A. x F/ b ph /\ y = b )
15		elequ2 1601	. . . . . . 7 |- ( y = b -> ( x e. y <-> x e. b ) )
16	15	adantl 262	. . . . . 6 |- ( ( A. x F/ b ph /\ y = b ) -> ( x e. y <-> x e. b ) )
17	16	bibi1d 222	. . . . 5 |- ( ( A. x F/ b ph /\ y = b ) -> ( ( x e. y <-> ( x e. a /\ ph ) ) <-> ( x e. b <-> ( x e. a /\ ph ) ) ) )
18	14, 17	albid 1506	. . . 4 |- ( ( A. x F/ b ph /\ 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/ b ph -> ( 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 1802	. 2 |- ( A. x F/ b ph -> ( E. y A. x ( x e. y <-> ( x e. a /\ ph ) ) <-> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) ) )
21	2, 20	mpbii 136	1 |- ( A. x F/ b ph -> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381  BOUNDED wbd 9932

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036

This theorem is referenced by:  bdsepnf  10008