Theorem bdsepnft 13769

Description: Closed form of bdsepnf 13770. Version of ax-bdsep 13766 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 13767 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 13768	. 2 |- E. y A. x ( x e. y <-> ( x e. a /\ ph ) )
3		nfnf1 1532	. . . 4 |- F/ b F/ b ph
4	3	nfal 1564	. . 3 |- F/ b A. x F/ b ph
5		nfa1 1529	. . . 4 |- F/ x A. x F/ b ph
6		nfvd 1517	. . . . 5 |- ( A. x F/ b ph -> F/ b x e. y )
7		nfv 1516	. . . . . . 7 |- F/ b x e. a
8	7	a1i 9	. . . . . 6 |- ( A. x F/ b ph -> F/ b x e. a )
9		sp 1499	. . . . . 6 |- ( A. x F/ b ph -> F/ b ph )
10	8, 9	nfand 1556	. . . . 5 |- ( A. x F/ b ph -> F/ b ( x e. a /\ ph ) )
11	6, 10	nfbid 1576	. . . 4 |- ( A. x F/ b ph -> F/ b ( x e. y <-> ( x e. a /\ ph ) ) )
12	5, 11	nfald 1748	. . 3 |- ( A. x F/ b ph -> F/ b A. x ( x e. y <-> ( x e. a /\ ph ) ) )
13		nfv 1516	. . . . . 6 |- F/ x y = b
14	5, 13	nfan 1553	. . . . 5 |- F/ x ( A. x F/ b ph /\ y = b )
15		elequ2 2141	. . . . . . 7 |- ( y = b -> ( x e. y <-> x e. b ) )
16	15	adantl 275	. . . . . 6 |- ( ( A. x F/ b ph /\ y = b ) -> ( x e. y <-> x e. b ) )
17	16	bibi1d 232	. . . . 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 1603	. . . 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 114	. . 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 1915	. 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 147	1 |- ( A. x F/ b ph -> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   F/wnf 1448   E.wex 1480  BOUNDED wbd 13694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161

This theorem is referenced by:  bdsepnf  13770