Theorem bdsepnfALT 9344

Description: Alternate proof of bdsepnf 9343, not using bdsepnft 9342. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bdsepnfALT	|- E. bA.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 9341	. 2 |- E.yA.x(x e. y <-> (x e. a /\ ph))
3		nfv 1418	. . . . 5 |- F/ b x e. y
4		nfv 1418	. . . . . 6 |- F/ b x e. a
5		bdsepnf.nf	. . . . . 6 |- F/ bph
6	4, 5	nfan 1454	. . . . 5 |- F/ b(x e. a /\ ph)
7	3, 6	nfbi 1478	. . . 4 |- F/ b(x e. y <-> (x e. a /\ ph))
8	7	nfal 1465	. . 3 |- F/ bA.x(x e. y <-> (x e. a /\ ph))
9		nfv 1418	. . 3 |- F/yA.x(x e. b <-> (x e. a /\ ph))
10		elequ2 1598	. . . . 5 |- (y = b -> (x e. y <-> x e. b))
11	10	bibi1d 222	. . . 4 |- (y = b -> ((x e. y <-> (x e. a /\ ph)) <-> (x e. b <-> (x e. a /\ ph))))
12	11	albidv 1702	. . 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 1636	. 2 |- (E.yA.x(x e. y <-> (x e. a /\ ph)) <-> E. bA.x(x e. b <-> (x e. a /\ ph)))
14	2, 13	mpbi 133	1 |- E. bA.x(x e. b <-> (x e. a /\ ph))

Syntax hints:   /\ wa 97   <-> wb 98  A.wal 1240   F/wnf 1346  E.wex 1378  BOUNDED wbd 9267

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bdsep 9339

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-cleq 2030  df-clel 2033

This theorem is referenced by: (None)