Theorem bdsepnfALT 14501

Description: Alternate proof of bdsepnf 14500, not using bdsepnft 14499. (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 14498	. 2 |- E. y A. x ( x e. y <-> ( x e. a /\ ph ) )
3		nfv 1528	. . . . 5 |- F/ b x e. y
4		nfv 1528	. . . . . 6 |- F/ b x e. a
5		bdsepnf.nf	. . . . . 6 |- F/ b ph
6	4, 5	nfan 1565	. . . . 5 |- F/ b ( x e. a /\ ph )
7	3, 6	nfbi 1589	. . . 4 |- F/ b ( x e. y <-> ( x e. a /\ ph ) )
8	7	nfal 1576	. . 3 |- F/ b A. x ( x e. y <-> ( x e. a /\ ph ) )
9		nfv 1528	. . 3 |- F/ y A. x ( x e. b <-> ( x e. a /\ ph ) )
10		elequ2 2153	. . . . 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 1824	. . 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 1756	. 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 1351   F/wnf 1460   E.wex 1492  BOUNDED wbd 14424

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-bdsep 14496

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-cleq 2170  df-clel 2173

This theorem is referenced by: (None)