Theorem strcollnfALT 9446

Description: Alternate proof of strcollnf 9445, not using strcollnft 9444. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
strcollnf.nf	|- F/ bph

Assertion

Ref	Expression
strcollnfALT	|- (A.x e. a E.yph -> E. bA.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 9443	. 2 |- (A.x e. a E.yph -> E.zA.y(y e. z <-> E.x e. a ph))
2		nfv 1418	. . . . 5 |- F/ b y e. z
3		nfcv 2175	. . . . . 6 |- F/_ b a
4		strcollnf.nf	. . . . . 6 |- F/ bph
5	3, 4	nfrexxy 2355	. . . . 5 |- F/ bE.x e. a ph
6	2, 5	nfbi 1478	. . . 4 |- F/ b(y e. z <-> E.x e. a ph)
7	6	nfal 1465	. . 3 |- F/ bA.y(y e. z <-> E.x e. a ph)
8		nfv 1418	. . 3 |- F/zA.y(y e. b <-> E.x e. a ph)
9		elequ2 1598	. . . . 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 1702	. . 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 1636	. 2 |- (E.zA.y(y e. z <-> E.x e. a ph) <-> E. bA.y(y e. b <-> E.x e. a ph))
13	1, 12	sylib 127	1 |- (A.x e. a E.yph -> E. bA.y(y e. b <-> E.x e. a ph))

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   F/wnf 1346  E.wex 1378  A.wral 2300  E.wrex 2301

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-strcoll 9442

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306

This theorem is referenced by: (None)