Theorem kmlem6 3585

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1.

Assertion

Ref	Expression
kmlem6	|- ((A.z e. x -. z = (/) /\ A.z e. x A.w e. x (ph -> A = (/))) -> A.z e. x E.v e. z A.w e. x (ph -> -. v e. A))

Distinct variable group(s):   x,z,w,v   ph,v   v,A

Proof of Theorem kmlem6

Step	Hyp	Ref	Expression
1		r19.26 1289	. 2 |- (A.z e. x (-. z = (/) /\ A.w e. x (ph -> A = (/))) <-> (A.z e. x -. z = (/) /\ A.z e. x A.w e. x (ph -> A = (/))))
2		19.29r 753	. . . . 5 |- ((E.v v e. z /\ A.vA.w e. x (ph -> -. v e. A)) -> E.v(v e. z /\ A.w e. x (ph -> -. v e. A)))
3		df-rex 1206	. . . . 5 |- (E.v e. z A.w e. x (ph -> -. v e. A) <-> E.v(v e. z /\ A.w e. x (ph -> -. v e. A)))
4	2, 3	sylibr 175	. . . 4 |- ((E.v v e. z /\ A.vA.w e. x (ph -> -. v e. A)) -> E.v e. z A.w e. x (ph -> -. v e. A))
5		n0 1714	. . . . 5 |- (-. z = (/) <-> E.v v e. z)
6	5	biimp 133	. . . 4 |- (-. z = (/) -> E.v v e. z)
7		n0i 1712	. . . . . . . 8 |- (v e. A -> -. A = (/))
8	7	con2i 89	. . . . . . 7 |- (A = (/) -> -. v e. A)
9	8	syl3 18	. . . . . 6 |- ((ph -> A = (/)) -> (ph -> -. v e. A))
10	9	r19.20si 1254	. . . . 5 |- (A.w e. x (ph -> A = (/)) -> A.w e. x (ph -> -. v e. A))
11	10	19.21aiv 943	. . . 4 |- (A.w e. x (ph -> A = (/)) -> A.vA.w e. x (ph -> -. v e. A))
12	4, 6, 11	syl2an 349	. . 3 |- ((-. z = (/) /\ A.w e. x (ph -> A = (/))) -> E.v e. z A.w e. x (ph -> -. v e. A))
13	12	r19.20si 1254	. 2 |- (A.z e. x (-. z = (/) /\ A.w e. x (ph -> A = (/))) -> A.z e. x E.v e. z A.w e. x (ph -> -. v e. A))
14	1, 13	sylbir 176	1 |- ((A.z e. x -. z = (/) /\ A.z e. x A.w e. x (ph -> A = (/))) -> A.z e. x E.v e. z A.w e. x (ph -> -. v e. A))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  (/)c0 1707

This theorem is referenced by:  kmlem7 3586

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-nul 1708

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