Theorem kmlem6 4780

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 1753	. 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 1074	. . . . 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 1653	. . . . 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 2292	. . . . 5 |- (z =/= (/) <-> E.v v e. z)
6	5	biimp 151	. . . 4 |- (z =/= (/) -> E.v v e. z)
7		ne0i 2289	. . . . . . . 8 |- (v e. A -> A =/= (/))
8	7	necon2bi 1615	. . . . . . 7 |- (A = (/) -> -. v e. A)
9	8	imim2i 17	. . . . . 6 |- ((ph -> A = (/)) -> (ph -> -. v e. A))
10	9	r19.20si 1709	. . . . 5 |- (A.w e. x (ph -> A = (/)) -> A.w e. x (ph -> -. v e. A))
11	10	19.21aiv 1288	. . . 4 |- (A.w e. x (ph -> A = (/)) -> A.vA.w e. x (ph -> -. v e. A))
12	4, 6, 11	syl2an 456	. . 3 |- ((z =/= (/) /\ A.w e. x (ph -> A = (/))) -> E.v e. z A.w e. x (ph -> -. v e. A))
13	12	r19.20si 1709	. 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 956   = wceq 958   e. wcel 960  E.wex 982   =/= wne 1588  A.wral 1648  E.wrex 1649  (/)c0 2283

This theorem is referenced by:  kmlem7 4781

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-nul 2284

Copyright terms: Public domain