Theorem bdsepnft 15092

Description: Closed form of bdsepnf 15093. Version of ax-bdsep 15089 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 15090 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 15091	. 2 |- E. y A. x ( x e. y <-> ( x e. a /\ ph ) )
3		nfnf1 1555	. . . 4 |- F/ b F/ b ph
4	3	nfal 1587	. . 3 |- F/ b A. x F/ b ph
5		nfa1 1552	. . . 4 |- F/ x A. x F/ b ph
6		nfvd 1540	. . . . 5 |- ( A. x F/ b ph -> F/ b x e. y )
7		nfv 1539	. . . . . . 7 |- F/ b x e. a
8	7	a1i 9	. . . . . 6 |- ( A. x F/ b ph -> F/ b x e. a )
9		sp 1522	. . . . . 6 |- ( A. x F/ b ph -> F/ b ph )
10	8, 9	nfand 1579	. . . . 5 |- ( A. x F/ b ph -> F/ b ( x e. a /\ ph ) )
11	6, 10	nfbid 1599	. . . 4 |- ( A. x F/ b ph -> F/ b ( x e. y <-> ( x e. a /\ ph ) ) )
12	5, 11	nfald 1771	. . 3 |- ( A. x F/ b ph -> F/ b A. x ( x e. y <-> ( x e. a /\ ph ) ) )
13		nfv 1539	. . . . . 6 |- F/ x y = b
14	5, 13	nfan 1576	. . . . 5 |- F/ x ( A. x F/ b ph /\ y = b )
15		elequ2 2165	. . . . . . 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 1626	. . . 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 1939	. 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 1362   F/wnf 1471   E.wex 1503  BOUNDED wbd 15017

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-bdsep 15089

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2182  df-clel 2185

This theorem is referenced by:  bdsepnf  15093