Theorem axcnvprim 4092

Description: ax-cnv 4081 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
axcnvprim	|- E.yA.zA.w(E.a(A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))) /\ a e. y) <-> E.e(A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))) /\ e e. x))

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

Proof of Theorem axcnvprim

Step	Hyp	Ref	Expression
1		ax-cnv 4081	. 2 |- E.yA.zA.w(<<z, w>> e. y <-> <<w, z>> e. x)
2		df-clel 2349	. . . . . 6 |- (<<z, w>> e. y <-> E.a(a = <<z, w>> /\ a e. y))
3		axprimlem2 4090	. . . . . . . 8 |- (a = <<z, w>> <-> A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))))
4	3	anbi1i 676	. . . . . . 7 |- ((a = <<z, w>> /\ a e. y) <-> (A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))) /\ a e. y))
5	4	exbii 1582	. . . . . 6 |- (E.a(a = <<z, w>> /\ a e. y) <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))) /\ a e. y))
6	2, 5	bitri 240	. . . . 5 |- (<<z, w>> e. y <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))) /\ a e. y))
7		df-clel 2349	. . . . . 6 |- (<<w, z>> e. x <-> E.e(e = <<w, z>> /\ e e. x))
8		axprimlem2 4090	. . . . . . . 8 |- (e = <<w, z>> <-> A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))))
9	8	anbi1i 676	. . . . . . 7 |- ((e = <<w, z>> /\ e e. x) <-> (A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))) /\ e e. x))
10	9	exbii 1582	. . . . . 6 |- (E.e(e = <<w, z>> /\ e e. x) <-> E.e(A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))) /\ e e. x))
11	7, 10	bitri 240	. . . . 5 |- (<<w, z>> e. x <-> E.e(A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))) /\ e e. x))
12	6, 11	bibi12i 306	. . . 4 |- ((<<z, w>> e. y <-> <<w, z>> e. x) <-> (E.a(A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))) /\ a e. y) <-> E.e(A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))) /\ e e. x)))
13	12	2albii 1567	. . 3 |- (A.zA.w(<<z, w>> e. y <-> <<w, z>> e. x) <-> A.zA.w(E.a(A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))) /\ a e. y) <-> E.e(A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))) /\ e e. x)))
14	13	exbii 1582	. 2 |- (E.yA.zA.w(<<z, w>> e. y <-> <<w, z>> e. x) <-> E.yA.zA.w(E.a(A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))) /\ a e. y) <-> E.e(A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))) /\ e e. x)))
15	1, 14	mpbi 199	1 |- E.yA.zA.w(E.a(A.b(b e. a <-> (A.c(c e. b <-> c = z) \/ A.d(d e. b <-> (d = z \/ d = w)))) /\ a e. y) <-> E.e(A.f(f e. e <-> (A.g(g e. f <-> g = w) \/ A.h(h e. f <-> (h = w \/ h = z)))) /\ e e. x))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  <<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-cnv 4081

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)