Theorem axssetprim 4093

Description: ax-sset 4083 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

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

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

Proof of Theorem axssetprim

Step	Hyp	Ref	Expression
1		ax-sset 4083	. 2 |- E.xA.yA.z(<<y, z>> e. x <-> A.e(e e. y -> e e. z))
2		df-clel 2349	. . . . . 6 |- (<<y, z>> e. x <-> E.a(a = <<y, z>> /\ a e. x))
3		axprimlem2 4090	. . . . . . . 8 |- (a = <<y, z>> <-> A.b(b e. a <-> (A.c(c e. b <-> c = y) \/ A.d(d e. b <-> (d = y \/ d = z)))))
4	3	anbi1i 676	. . . . . . 7 |- ((a = <<y, z>> /\ a e. x) <-> (A.b(b e. a <-> (A.c(c e. b <-> c = y) \/ A.d(d e. b <-> (d = y \/ d = z)))) /\ a e. x))
5	4	exbii 1582	. . . . . 6 |- (E.a(a = <<y, z>> /\ a e. x) <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> c = y) \/ A.d(d e. b <-> (d = y \/ d = z)))) /\ a e. x))
6	2, 5	bitri 240	. . . . 5 |- (<<y, z>> e. x <-> E.a(A.b(b e. a <-> (A.c(c e. b <-> c = y) \/ A.d(d e. b <-> (d = y \/ d = z)))) /\ a e. x))
7	6	bibi1i 305	. . . 4 |- ((<<y, z>> e. x <-> A.e(e e. y -> e e. z)) <-> (E.a(A.b(b e. a <-> (A.c(c e. b <-> c = y) \/ A.d(d e. b <-> (d = y \/ d = z)))) /\ a e. x) <-> A.e(e e. y -> e e. z)))
8	7	2albii 1567	. . 3 |- (A.yA.z(<<y, z>> e. x <-> A.e(e e. y -> e e. z)) <-> A.yA.z(E.a(A.b(b e. a <-> (A.c(c e. b <-> c = y) \/ A.d(d e. b <-> (d = y \/ d = z)))) /\ a e. x) <-> A.e(e e. y -> e e. z)))
9	8	exbii 1582	. 2 |- (E.xA.yA.z(<<y, z>> e. x <-> A.e(e e. y -> e e. z)) <-> E.xA.yA.z(E.a(A.b(b e. a <-> (A.c(c e. b <-> c = y) \/ A.d(d e. b <-> (d = y \/ d = z)))) /\ a e. x) <-> A.e(e e. y -> e e. z)))
10	1, 9	mpbi 199	1 |- E.xA.yA.z(E.a(A.b(b e. a <-> (A.c(c e. b <-> c = y) \/ A.d(d e. b <-> (d = y \/ d = z)))) /\ a e. x) <-> A.e(e e. y -> e e. z))

Syntax hints:   -> wi 4   <-> 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-sset 4083

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)