Theorem strcollnfALT 10111

Description: Alternate proof of strcollnf 10110, not using strcollnft 10109. (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 10108	. 2 |- ( A. x e. a E. y ph -> E. z A. y ( y e. z <-> E. x e. a ph ) )
2		nfv 1421	. . . . 5 |- F/ b y e. z
3		nfcv 2178	. . . . . 6 |- F/_ b a
4		strcollnf.nf	. . . . . 6 |- F/ b ph
5	3, 4	nfrexxy 2361	. . . . 5 |- F/ b E. x e. a ph
6	2, 5	nfbi 1481	. . . 4 |- F/ b ( y e. z <-> E. x e. a ph )
7	6	nfal 1468	. . 3 |- F/ b A. y ( y e. z <-> E. x e. a ph )
8		nfv 1421	. . 3 |- F/ z A. y ( y e. b <-> E. x e. a ph )
9		elequ2 1601	. . . . 5 |- ( z = b -> ( y e. z <-> y e. b ) )
10	9	bibi1d 222	. . . 4 |- ( z = b -> ( ( y e. z <-> E. x e. a ph ) <-> ( y e. b <-> E. x e. a ph ) ) )
11	10	albidv 1705	. . 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 1639	. 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 127	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 98   A.wal 1241   F/wnf 1349   E.wex 1381   A.wral 2306   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-strcoll 10107

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312

This theorem is referenced by: (None)