Theorem strcollnfALT 12154

Description: Alternate proof of strcollnf 12153, not using strcollnft 12152. (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 12151	. 2 |- ( A. x e. a E. y ph -> E. z A. y ( y e. z <-> E. x e. a ph ) )
2		nfv 1467	. . . . 5 |- F/ b y e. z
3		nfcv 2229	. . . . . 6 |- F/_ b a
4		strcollnf.nf	. . . . . 6 |- F/ b ph
5	3, 4	nfrexxy 2416	. . . . 5 |- F/ b E. x e. a ph
6	2, 5	nfbi 1527	. . . 4 |- F/ b ( y e. z <-> E. x e. a ph )
7	6	nfal 1514	. . 3 |- F/ b A. y ( y e. z <-> E. x e. a ph )
8		nfv 1467	. . 3 |- F/ z A. y ( y e. b <-> E. x e. a ph )
9		elequ2 1649	. . . . 5 |- ( z = b -> ( y e. z <-> y e. b ) )
10	9	bibi1d 232	. . . 4 |- ( z = b -> ( ( y e. z <-> E. x e. a ph ) <-> ( y e. b <-> E. x e. a ph ) ) )
11	10	albidv 1753	. . 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 1687	. 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 121	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 104   A.wal 1288   F/wnf 1395   E.wex 1427   A.wral 2360   E.wrex 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-strcoll 12150

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366

This theorem is referenced by: (None)