Theorem kmlem6 4770

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 groups:   v,A   x,v,ph   w,v,z,x

Proof of Theorem kmlem6

Step	Hyp	Ref	Expression
1		r19.26 1750	. 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 1072	. . . . 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 1650	. . . . 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 200	. . . 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		ne0 2288	. . . . 5 |- (z =/= (/) <-> E.v v e. z)
6	5	biimp 151	. . . 4 |- (z =/= (/) -> E.v v e. z)
7		ne0i 2286	. . . . . . . 8 |- (v e. A -> A =/= (/))
8	7	necon2bi 1612	. . . . . . 7 |- (A = (/) -> -. v e. A)
9	8	imim2i 17	. . . . . 6 |- ((ph -> A = (/)) -> (ph -> -. v e. A))
10	9	r19.20si 1706	. . . . 5 |- (A.w e. x (ph -> A = (/)) -> A.w e. x (ph -> -. v e. A))
11	10	19.21aiv 1286	. . . 4 |- (A.w e. x (ph -> A = (/)) -> A.vA.w e. x (ph -> -. v e. A))
12	4, 6, 11	syl2an 454	. . 3 |- ((z =/= (/) /\ A.w e. x (ph -> A = (/))) -> E.v e. z A.w e. x (ph -> -. v e. A))
13	12	r19.20si 1706	. 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 201	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 2   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  E.wrex 1646  (/)c0 2280

This theorem is referenced by:  kmlem7 4771

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-nul 2281

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