Theorem strcollnfALT 11527

Description: Alternate proof of strcollnf 11526, not using strcollnft 11525. (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 11524	. 2 |- ( A. x e. a E. y ph -> E. z A. y ( y e. z <-> E. x e. a ph ) )
2		nfv 1466	. . . . 5 |- F/ b y e. z
3		nfcv 2228	. . . . . 6 |- F/_ b a
4		strcollnf.nf	. . . . . 6 |- F/ b ph
5	3, 4	nfrexxy 2415	. . . . 5 |- F/ b E. x e. a ph
6	2, 5	nfbi 1526	. . . 4 |- F/ b ( y e. z <-> E. x e. a ph )
7	6	nfal 1513	. . 3 |- F/ b A. y ( y e. z <-> E. x e. a ph )
8		nfv 1466	. . 3 |- F/ z A. y ( y e. b <-> E. x e. a ph )
9		elequ2 1648	. . . . 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 1752	. . 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 1686	. 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 1287   F/wnf 1394   E.wex 1426   A.wral 2359   E.wrex 2360

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-strcoll 11523

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365

This theorem is referenced by: (None)