Theorem sscoll2 9448

Description: Version of ax-sscoll 9447 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)

Assertion

Ref	Expression
sscoll2	|- E. cA.z(A.x e. a E.y e. b ph -> E. d e. c A.y(y e. d <-> E.x e. a ph))

Distinct variable groups:   a, b, c, d,x,y,z   ph, c, d

Allowed substitution hints:   ph(x,y,z, a, b)

Proof of Theorem sscoll2

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1418	. . 3 |- F/ c(u = a /\ v = b)
2		nfv 1418	. . . 4 |- F/z(u = a /\ v = b)
3		simpl 102	. . . . . 6 |- ((u = a /\ v = b) -> u = a)
4		rexeq 2500	. . . . . . 7 |- (v = b -> (E.y e. v ph <-> E.y e. b ph))
5	4	adantl 262	. . . . . 6 |- ((u = a /\ v = b) -> (E.y e. v ph <-> E.y e. b ph))
6	3, 5	raleqbidv 2511	. . . . 5 |- ((u = a /\ v = b) -> (A.x e. u E.y e. v ph <-> A.x e. a E.y e. b ph))
7		nfv 1418	. . . . . 6 |- F/ d(u = a /\ v = b)
8		nfv 1418	. . . . . . 7 |- F/y(u = a /\ v = b)
9		rexeq 2500	. . . . . . . . 9 |- (u = a -> (E.x e. u ph <-> E.x e. a ph))
10	9	adantr 261	. . . . . . . 8 |- ((u = a /\ v = b) -> (E.x e. u ph <-> E.x e. a ph))
11	10	bibi2d 221	. . . . . . 7 |- ((u = a /\ v = b) -> ((y e. d <-> E.x e. u ph) <-> (y e. d <-> E.x e. a ph)))
12	8, 11	albid 1503	. . . . . 6 |- ((u = a /\ v = b) -> (A.y(y e. d <-> E.x e. u ph) <-> A.y(y e. d <-> E.x e. a ph)))
13	7, 12	rexbid 2319	. . . . 5 |- ((u = a /\ v = b) -> (E. d e. c A.y(y e. d <-> E.x e. u ph) <-> E. d e. c A.y(y e. d <-> E.x e. a ph)))
14	6, 13	imbi12d 223	. . . 4 |- ((u = a /\ v = b) -> ((A.x e. u E.y e. v ph -> E. d e. c A.y(y e. d <-> E.x e. u ph)) <-> (A.x e. a E.y e. b ph -> E. d e. c A.y(y e. d <-> E.x e. a ph))))
15	2, 14	albid 1503	. . 3 |- ((u = a /\ v = b) -> (A.z(A.x e. u E.y e. v ph -> E. d e. c A.y(y e. d <-> E.x e. u ph)) <-> A.z(A.x e. a E.y e. b ph -> E. d e. c A.y(y e. d <-> E.x e. a ph))))
16	1, 15	exbid 1504	. 2 |- ((u = a /\ v = b) -> (E. cA.z(A.x e. u E.y e. v ph -> E. d e. c A.y(y e. d <-> E.x e. u ph)) <-> E. cA.z(A.x e. a E.y e. b ph -> E. d e. c A.y(y e. d <-> E.x e. a ph))))
17		ax-sscoll 9447	. . . 4 |- A.uA.vE. cA.z(A.x e. u E.y e. v ph -> E. d e. c A.y(y e. d <-> E.x e. u ph))
18	17	spi 1426	. . 3 |- A.vE. cA.z(A.x e. u E.y e. v ph -> E. d e. c A.y(y e. d <-> E.x e. u ph))
19	18	spi 1426	. 2 |- E. cA.z(A.x e. u E.y e. v ph -> E. d e. c A.y(y e. d <-> E.x e. u ph))
20	16, 19	ch2varv 9243	1 |- E. cA.z(A.x e. a E.y e. b ph -> E. d e. c A.y(y e. d <-> E.x e. a ph))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sscoll 9447

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)