Theorem kmlem1 5927

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2.

Assertion

Ref	Expression
kmlem1	|- (A.x((A.z e. x z =/= (/) /\ A.z e. x A.w e. x ph) -> E.yA.z e. x ps) -> A.x(A.z e. x A.w e. x ph -> E.yA.z e. x (z =/= (/) -> ps)))

Distinct variable groups:   x,y,ph   ps,x   x,w,y,z

Proof of Theorem kmlem1

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . 6 |- v e. _V
2	1	rabex 3461	. . . . 5 |- {u e. v | u =/= (/)} e. _V
3		raleq 2266	. . . . . . 7 |- (x = {u e. v | u =/= (/)} -> (A.z e. x z =/= (/) <-> A.z e. {u e. v | u =/= (/)}z =/= (/)))
4		raleq 2266	. . . . . . . 8 |- (x = {u e. v | u =/= (/)} -> (A.w e. x ph <-> A.w e. {u e. v | u =/= (/)}ph))
5	4	raleqbi1dv 2271	. . . . . . 7 |- (x = {u e. v | u =/= (/)} -> (A.z e. x A.w e. x ph <-> A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph))
6	3, 5	anbi12d 690	. . . . . 6 |- (x = {u e. v | u =/= (/)} -> ((A.z e. x z =/= (/) /\ A.z e. x A.w e. x ph) <-> (A.z e. {u e. v | u =/= (/)}z =/= (/) /\ A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph)))
7		raleq 2266	. . . . . . 7 |- (x = {u e. v | u =/= (/)} -> (A.z e. x ps <-> A.z e. {u e. v | u =/= (/)}ps))
8	7	exbidv 1657	. . . . . 6 |- (x = {u e. v | u =/= (/)} -> (E.yA.z e. x ps <-> E.yA.z e. {u e. v | u =/= (/)}ps))
9	6, 8	imbi12d 688	. . . . 5 |- (x = {u e. v | u =/= (/)} -> (((A.z e. x z =/= (/) /\ A.z e. x A.w e. x ph) -> E.yA.z e. x ps) <-> ((A.z e. {u e. v | u =/= (/)}z =/= (/) /\ A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph) -> E.yA.z e. {u e. v | u =/= (/)}ps)))
10	2, 9	cla4v 2370	. . . 4 |- (A.x((A.z e. x z =/= (/) /\ A.z e. x A.w e. x ph) -> E.yA.z e. x ps) -> ((A.z e. {u e. v | u =/= (/)}z =/= (/) /\ A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph) -> E.yA.z e. {u e. v | u =/= (/)}ps))
11	10	19.21aiv 1664	. . 3 |- (A.x((A.z e. x z =/= (/) /\ A.z e. x A.w e. x ph) -> E.yA.z e. x ps) -> A.v((A.z e. {u e. v | u =/= (/)}z =/= (/) /\ A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph) -> E.yA.z e. {u e. v | u =/= (/)}ps))
12		ssrab2 2692	. . . . . . . . 9 |- {u e. v | u =/= (/)} C_ v
13	12	sseli 2617	. . . . . . . 8 |- (z e. {u e. v | u =/= (/)} -> z e. v)
14	12	sseli 2617	. . . . . . . . . 10 |- (w e. {u e. v | u =/= (/)} -> w e. v)
15	14	imim1i 19	. . . . . . . . 9 |- ((w e. v -> ph) -> (w e. {u e. v | u =/= (/)} -> ph))
16	15	ralimi2 2165	. . . . . . . 8 |- (A.w e. v ph -> A.w e. {u e. v | u =/= (/)}ph)
17	13, 16	imim12i 21	. . . . . . 7 |- ((z e. v -> A.w e. v ph) -> (z e. {u e. v | u =/= (/)} -> A.w e. {u e. v | u =/= (/)}ph))
18	17	ralimi2 2165	. . . . . 6 |- (A.z e. v A.w e. v ph -> A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph)
19		neeq1 2024	. . . . . . . . 9 |- (u = z -> (u =/= (/) <-> z =/= (/)))
20	19	elrab 2414	. . . . . . . 8 |- (z e. {u e. v | u =/= (/)} <-> (z e. v /\ z =/= (/)))
21	20	simprbi 353	. . . . . . 7 |- (z e. {u e. v | u =/= (/)} -> z =/= (/))
22	21	rgen 2159	. . . . . 6 |- A.z e. {u e. v | u =/= (/)}z =/= (/)
23	18, 22	jctil 316	. . . . 5 |- (A.z e. v A.w e. v ph -> (A.z e. {u e. v | u =/= (/)}z =/= (/) /\ A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph))
24	20	biimpri 169	. . . . . . . . 9 |- ((z e. v /\ z =/= (/)) -> z e. {u e. v | u =/= (/)})
25	24	imim1i 19	. . . . . . . 8 |- ((z e. {u e. v | u =/= (/)} -> ps) -> ((z e. v /\ z =/= (/)) -> ps))
26	25	exp3a 405	. . . . . . 7 |- ((z e. {u e. v | u =/= (/)} -> ps) -> (z e. v -> (z =/= (/) -> ps)))
27	26	ralimi2 2165	. . . . . 6 |- (A.z e. {u e. v | u =/= (/)}ps -> A.z e. v (z =/= (/) -> ps))
28	27	eximi 1387	. . . . 5 |- (E.yA.z e. {u e. v | u =/= (/)}ps -> E.yA.z e. v (z =/= (/) -> ps))
29	23, 28	imim12i 21	. . . 4 |- (((A.z e. {u e. v | u =/= (/)}z =/= (/) /\ A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph) -> E.yA.z e. {u e. v | u =/= (/)}ps) -> (A.z e. v A.w e. v ph -> E.yA.z e. v (z =/= (/) -> ps)))
30	29	alimi 1338	. . 3 |- (A.v((A.z e. {u e. v | u =/= (/)}z =/= (/) /\ A.z e. {u e. v | u =/= (/)}A.w e. {u e. v | u =/= (/)}ph) -> E.yA.z e. {u e. v | u =/= (/)}ps) -> A.v(A.z e. v A.w e. v ph -> E.yA.z e. v (z =/= (/) -> ps)))
31	11, 30	syl 12	. 2 |- (A.x((A.z e. x z =/= (/) /\ A.z e. x A.w e. x ph) -> E.yA.z e. x ps) -> A.v(A.z e. v A.w e. v ph -> E.yA.z e. v (z =/= (/) -> ps)))
32		raleq 2266	. . . . 5 |- (v = x -> (A.w e. v ph <-> A.w e. x ph))
33	32	raleqbi1dv 2271	. . . 4 |- (v = x -> (A.z e. v A.w e. v ph <-> A.z e. x A.w e. x ph))
34		raleq 2266	. . . . 5 |- (v = x -> (A.z e. v (z =/= (/) -> ps) <-> A.z e. x (z =/= (/) -> ps)))
35	34	exbidv 1657	. . . 4 |- (v = x -> (E.yA.z e. v (z =/= (/) -> ps) <-> E.yA.z e. x (z =/= (/) -> ps)))
36	33, 35	imbi12d 688	. . 3 |- (v = x -> ((A.z e. v A.w e. v ph -> E.yA.z e. v (z =/= (/) -> ps)) <-> (A.z e. x A.w e. x ph -> E.yA.z e. x (z =/= (/) -> ps))))
37	36	cbvalv 1696	. 2 |- (A.v(A.z e. v A.w e. v ph -> E.yA.z e. v (z =/= (/) -> ps)) <-> A.x(A.z e. x A.w e. x ph -> E.yA.z e. x (z =/= (/) -> ps)))
38	31, 37	sylib 215	1 |- (A.x((A.z e. x z =/= (/) /\ A.z e. x A.w e. x ph) -> E.yA.z e. x ps) -> A.x(A.z e. x A.w e. x ph -> E.yA.z e. x (z =/= (/) -> ps)))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  {crab 2108  (/)c0 2875

This theorem is referenced by:  kmlem13 5939

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rab 2112  df-v 2294  df-in 2603  df-ss 2605

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