Theorem strcollnfALT 10781

Description: Alternate proof of strcollnf 10780, not using strcollnft 10779. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
strcollnf.nf	|- F/ b ph

Assertion

Ref	Expression
strcollnfALT	|- ( A. x e. a E. y ph -> E. b A. y ( y e. b <-> E. x e. a ph ) )

Distinct variable group:   a, b, x, y

Allowed substitution hints:   ph( x, y, a, b)

Proof of Theorem strcollnfALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		strcoll2 10778	. 2 |- ( A. x e. a E. y ph -> E. z A. y ( y e. z <-> E. x e. a ph ) )
2		nfv 1461	. . . . 5 |- F/ b y e. z
3		nfcv 2219	. . . . . 6 |- F/_ b a
4		strcollnf.nf	. . . . . 6 |- F/ b ph
5	3, 4	nfrexxy 2403	. . . . 5 |- F/ b E. x e. a ph
6	2, 5	nfbi 1521	. . . 4 |- F/ b ( y e. z <-> E. x e. a ph )
7	6	nfal 1508	. . 3 |- F/ b A. y ( y e. z <-> E. x e. a ph )
8		nfv 1461	. . 3 |- F/ z A. y ( y e. b <-> E. x e. a ph )
9		elequ2 1641	. . . . 5 |- ( z = b -> ( y e. z <-> y e. b ) )
10	9	bibi1d 231	. . . 4 |- ( z = b -> ( ( y e. z <-> E. x e. a ph ) <-> ( y e. b <-> E. x e. a ph ) ) )
11	10	albidv 1745	. . 3 |- ( z = b -> ( A. y ( y e. z <-> E. x e. a ph ) <-> A. y ( y e. b <-> E. x e. a ph ) ) )
12	7, 8, 11	cbvex 1679	. 2 |- ( E. z A. y ( y e. z <-> E. x e. a ph ) <-> E. b A. y ( y e. b <-> E. x e. a ph ) )
13	1, 12	sylib 120	1 |- ( A. x e. a E. y ph -> E. b A. y ( y e. b <-> E. x e. a ph ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   F/wnf 1389   E.wex 1421   A.wral 2348   E.wrex 2349

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-strcoll 10777

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354

This theorem is referenced by: (None)