Theorem strcollnfALT 13173

Description: Alternate proof of strcollnf 13172, not using strcollnft 13171. (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 13170	. 2 |- ( A. x e. a E. y ph -> E. z A. y ( y e. z <-> E. x e. a ph ) )
2		nfv 1508	. . . . 5 |- F/ b y e. z
3		nfcv 2279	. . . . . 6 |- F/_ b a
4		strcollnf.nf	. . . . . 6 |- F/ b ph
5	3, 4	nfrexxy 2470	. . . . 5 |- F/ b E. x e. a ph
6	2, 5	nfbi 1568	. . . 4 |- F/ b ( y e. z <-> E. x e. a ph )
7	6	nfal 1555	. . 3 |- F/ b A. y ( y e. z <-> E. x e. a ph )
8		nfv 1508	. . 3 |- F/ z A. y ( y e. b <-> E. x e. a ph )
9		elequ2 1691	. . . . 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 1796	. . 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 1729	. 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 1329   F/wnf 1436   E.wex 1468   A.wral 2414   E.wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-strcoll 13169

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420

This theorem is referenced by: (None)