Theorem bdsepnfALT 12668

Description: Alternate proof of bdsepnf 12667, not using bdsepnft 12666. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bdsepnf.nf	|- F/ b ph
bdsepnf.1	|- BOUNDED ph

Assertion

Ref	Expression
bdsepnfALT	|- 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 bdsepnfALT

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdsepnf.1	. . 3 |- BOUNDED ph
2	1	bdsep2 12665	. 2 |- E. y A. x ( x e. y <-> ( x e. a /\ ph ) )
3		nfv 1476	. . . . 5 |- F/ b x e. y
4		nfv 1476	. . . . . 6 |- F/ b x e. a
5		bdsepnf.nf	. . . . . 6 |- F/ b ph
6	4, 5	nfan 1512	. . . . 5 |- F/ b ( x e. a /\ ph )
7	3, 6	nfbi 1536	. . . 4 |- F/ b ( x e. y <-> ( x e. a /\ ph ) )
8	7	nfal 1523	. . 3 |- F/ b A. x ( x e. y <-> ( x e. a /\ ph ) )
9		nfv 1476	. . 3 |- F/ y A. x ( x e. b <-> ( x e. a /\ ph ) )
10		elequ2 1659	. . . . 5 |- ( y = b -> ( x e. y <-> x e. b ) )
11	10	bibi1d 232	. . . 4 |- ( y = b -> ( ( x e. y <-> ( x e. a /\ ph ) ) <-> ( x e. b <-> ( x e. a /\ ph ) ) ) )
12	11	albidv 1763	. . 3 |- ( y = b -> ( A. x ( x e. y <-> ( x e. a /\ ph ) ) <-> A. x ( x e. b <-> ( x e. a /\ ph ) ) ) )
13	8, 9, 12	cbvex 1697	. 2 |- ( E. y A. x ( x e. y <-> ( x e. a /\ ph ) ) <-> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) )
14	2, 13	mpbi 144	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1297   F/wnf 1404   E.wex 1436  BOUNDED wbd 12591

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-bdsep 12663

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-cleq 2093  df-clel 2096

This theorem is referenced by: (None)