Theorem axxpprim 4091

Description: ax-xp 4080 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

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

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

Proof of Theorem axxpprim

Step	Hyp	Ref	Expression
1		ax-xp 4080	. 2 |- E.yA.z(z e. y <-> E.wE.t(z = <<w, t>> /\ t e. x))
2		axprimlem2 4090	. . . . . . 7 |- (z = <<w, t>> <-> A.a(a e. z <-> (A.b(b e. a <-> b = w) \/ A.c(c e. a <-> (c = w \/ c = t)))))
3	2	anbi1i 676	. . . . . 6 |- ((z = <<w, t>> /\ t e. x) <-> (A.a(a e. z <-> (A.b(b e. a <-> b = w) \/ A.c(c e. a <-> (c = w \/ c = t)))) /\ t e. x))
4	3	2exbii 1583	. . . . 5 |- (E.wE.t(z = <<w, t>> /\ t e. x) <-> E.wE.t(A.a(a e. z <-> (A.b(b e. a <-> b = w) \/ A.c(c e. a <-> (c = w \/ c = t)))) /\ t e. x))
5	4	bibi2i 304	. . . 4 |- ((z e. y <-> E.wE.t(z = <<w, t>> /\ t e. x)) <-> (z e. y <-> E.wE.t(A.a(a e. z <-> (A.b(b e. a <-> b = w) \/ A.c(c e. a <-> (c = w \/ c = t)))) /\ t e. x)))
6	5	albii 1566	. . 3 |- (A.z(z e. y <-> E.wE.t(z = <<w, t>> /\ t e. x)) <-> A.z(z e. y <-> E.wE.t(A.a(a e. z <-> (A.b(b e. a <-> b = w) \/ A.c(c e. a <-> (c = w \/ c = t)))) /\ t e. x)))
7	6	exbii 1582	. 2 |- (E.yA.z(z e. y <-> E.wE.t(z = <<w, t>> /\ t e. x)) <-> E.yA.z(z e. y <-> E.wE.t(A.a(a e. z <-> (A.b(b e. a <-> b = w) \/ A.c(c e. a <-> (c = w \/ c = t)))) /\ t e. x)))
8	1, 7	mpbi 199	1 |- E.yA.z(z e. y <-> E.wE.t(A.a(a e. z <-> (A.b(b e. a <-> b = w) \/ A.c(c e. a <-> (c = w \/ c = t)))) /\ t e. x))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  <<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-xp 4080

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)