Theorem axtyplowerprim 4095

Description: ax-typlower 4087 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
axtyplowerprim	|- E.yA.z(z e. y <-> A.wE.a(A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x))

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

Proof of Theorem axtyplowerprim

Step	Hyp	Ref	Expression
1		ax-typlower 4087	. 2 |- E.yA.z(z e. y <-> A.w<<w, {z}>> e. x)
2		df-clel 2349	. . . . . . 7 |- (<<w, {z}>> e. x <-> E.a(a = <<w, {z}>> /\ a e. x))
3		axprimlem2 4090	. . . . . . . . . 10 |- (a = <<w, {z}>> <-> A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ d = {z})))))
4		axprimlem1 4089	. . . . . . . . . . . . . . . 16 |- (d = {z} <-> A.e(e e. d <-> e = z))
5	4	orbi2i 505	. . . . . . . . . . . . . . 15 |- ((d = w \/ d = {z}) <-> (d = w \/ A.e(e e. d <-> e = z)))
6	5	bibi2i 304	. . . . . . . . . . . . . 14 |- ((d e. b <-> (d = w \/ d = {z})) <-> (d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))
7	6	albii 1566	. . . . . . . . . . . . 13 |- (A.d(d e. b <-> (d = w \/ d = {z})) <-> A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))
8	7	orbi2i 505	. . . . . . . . . . . 12 |- ((A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ d = {z}))) <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z)))))
9	8	bibi2i 304	. . . . . . . . . . 11 |- ((b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ d = {z})))) <-> (b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))))
10	9	albii 1566	. . . . . . . . . 10 |- (A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ d = {z})))) <-> A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))))
11	3, 10	bitri 240	. . . . . . . . 9 |- (a = <<w, {z}>> <-> A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))))
12	11	anbi1i 676	. . . . . . . 8 |- ((a = <<w, {z}>> /\ a e. x) <-> (A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x))
13	12	exbii 1582	. . . . . . 7 |- (E.a(a = <<w, {z}>> /\ a e. x) <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x))
14	2, 13	bitri 240	. . . . . 6 |- (<<w, {z}>> e. x <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x))
15	14	albii 1566	. . . . 5 |- (A.w<<w, {z}>> e. x <-> A.wE.a(A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x))
16	15	bibi2i 304	. . . 4 |- ((z e. y <-> A.w<<w, {z}>> e. x) <-> (z e. y <-> A.wE.a(A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x)))
17	16	albii 1566	. . 3 |- (A.z(z e. y <-> A.w<<w, {z}>> e. x) <-> A.z(z e. y <-> A.wE.a(A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x)))
18	17	exbii 1582	. 2 |- (E.yA.z(z e. y <-> A.w<<w, {z}>> e. x) <-> E.yA.z(z e. y <-> A.wE.a(A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x)))
19	1, 18	mpbi 199	1 |- E.yA.z(z e. y <-> A.wE.a(A.b(b e. a <-> (A.c(c e. b <-> c = w) \/ A.d(d e. b <-> (d = w \/ A.e(e e. d <-> e = z))))) /\ a e. x))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {csn 3738  <<copk 4058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-typlower 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by: (None)