Theorem bdsepnft 16250

Description: Closed form of bdsepnf 16251. Version of ax-bdsep 16247 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 16248 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 16249	. 2 |- E. y A. x ( x e. y <-> ( x e. a /\ ph ) )
3		nfnf1 1590	. . . 4 |- F/ b F/ b ph
4	3	nfal 1622	. . 3 |- F/ b A. x F/ b ph
5		nfa1 1587	. . . 4 |- F/ x A. x F/ b ph
6		nfvd 1575	. . . . 5 |- ( A. x F/ b ph -> F/ b x e. y )
7		nfv 1574	. . . . . . 7 |- F/ b x e. a
8	7	a1i 9	. . . . . 6 |- ( A. x F/ b ph -> F/ b x e. a )
9		sp 1557	. . . . . 6 |- ( A. x F/ b ph -> F/ b ph )
10	8, 9	nfand 1614	. . . . 5 |- ( A. x F/ b ph -> F/ b ( x e. a /\ ph ) )
11	6, 10	nfbid 1634	. . . 4 |- ( A. x F/ b ph -> F/ b ( x e. y <-> ( x e. a /\ ph ) ) )
12	5, 11	nfald 1806	. . 3 |- ( A. x F/ b ph -> F/ b A. x ( x e. y <-> ( x e. a /\ ph ) ) )
13		nfv 1574	. . . . . 6 |- F/ x y = b
14	5, 13	nfan 1611	. . . . 5 |- F/ x ( A. x F/ b ph /\ y = b )
15		elequ2 2205	. . . . . . 7 |- ( y = b -> ( x e. y <-> x e. b ) )
16	15	adantl 277	. . . . . 6 |- ( ( A. x F/ b ph /\ y = b ) -> ( x e. y <-> x e. b ) )
17	16	bibi1d 233	. . . . 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 1661	. . . 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 115	. . 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 1974	. 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 148	1 |- ( A. x F/ b ph -> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   F/wnf 1506   E.wex 1538  BOUNDED wbd 16175

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-bdsep 16247

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-cleq 2222  df-clel 2225

This theorem is referenced by:  bdsepnf  16251