Theorem bdsepnfALT 10947

Description: Alternate proof of bdsepnf 10946, not using bdsepnft 10945. (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 10944	. 2 |- E. y A. x ( x e. y <-> ( x e. a /\ ph ) )
3		nfv 1462	. . . . 5 |- F/ b x e. y
4		nfv 1462	. . . . . 6 |- F/ b x e. a
5		bdsepnf.nf	. . . . . 6 |- F/ b ph
6	4, 5	nfan 1498	. . . . 5 |- F/ b ( x e. a /\ ph )
7	3, 6	nfbi 1522	. . . 4 |- F/ b ( x e. y <-> ( x e. a /\ ph ) )
8	7	nfal 1509	. . 3 |- F/ b A. x ( x e. y <-> ( x e. a /\ ph ) )
9		nfv 1462	. . 3 |- F/ y A. x ( x e. b <-> ( x e. a /\ ph ) )
10		elequ2 1643	. . . . 5 |- ( y = b -> ( x e. y <-> x e. b ) )
11	10	bibi1d 231	. . . 4 |- ( y = b -> ( ( x e. y <-> ( x e. a /\ ph ) ) <-> ( x e. b <-> ( x e. a /\ ph ) ) ) )
12	11	albidv 1747	. . 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 1681	. 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 143	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   A.wal 1283   F/wnf 1390   E.wex 1422  BOUNDED wbd 10870

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bdsep 10942

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-cleq 2076  df-clel 2079

This theorem is referenced by: (None)