Theorem bdsepnfALT 15619

Description: Alternate proof of bdsepnf 15618, not using bdsepnft 15617. (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 15616	. 2 |- E. y A. x ( x e. y <-> ( x e. a /\ ph ) )
3		nfv 1542	. . . . 5 |- F/ b x e. y
4		nfv 1542	. . . . . 6 |- F/ b x e. a
5		bdsepnf.nf	. . . . . 6 |- F/ b ph
6	4, 5	nfan 1579	. . . . 5 |- F/ b ( x e. a /\ ph )
7	3, 6	nfbi 1603	. . . 4 |- F/ b ( x e. y <-> ( x e. a /\ ph ) )
8	7	nfal 1590	. . 3 |- F/ b A. x ( x e. y <-> ( x e. a /\ ph ) )
9		nfv 1542	. . 3 |- F/ y A. x ( x e. b <-> ( x e. a /\ ph ) )
10		elequ2 2172	. . . . 5 |- ( y = b -> ( x e. y <-> x e. b ) )
11	10	bibi1d 233	. . . 4 |- ( y = b -> ( ( x e. y <-> ( x e. a /\ ph ) ) <-> ( x e. b <-> ( x e. a /\ ph ) ) ) )
12	11	albidv 1838	. . 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 1770	. 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 145	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1362   F/wnf 1474   E.wex 1506  BOUNDED wbd 15542

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-bdsep 15614

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-cleq 2189  df-clel 2192

This theorem is referenced by: (None)