Theorem kmlem1 4689

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 1788	. . . . . 6 |- v e. V
2	1	rabex 2693	. . . . 5 |- {u e. v | u =/= (/)} e. V
3		raleq1 1762	. . . . . . 7 |- (x = {u e. v | u =/= (/)} -> (A.z e. x z =/= (/) <-> A.z e. {u e. v | u =/= (/)}z =/= (/)))
4		raleq1 1762	. . . . . . . 8 |- (x = {u e. v | u =/= (/)} -> (A.w e. x ph <-> A.w e. {u e. v | u =/= (/)}ph))
5	4	raleqd 1767	. . . . . . 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 626	. . . . . 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		raleq1 1762	. . . . . . 7 |- (x = {u e. v | u =/= (/)} -> (A.z e. x ps <-> A.z e. {u e. v | u =/= (/)}ps))
8	7	exbidv 1261	. . . . . 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 624	. . . . 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 1841	. . . 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 1268	. . 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 2102	. . . . . . . . 9 |- {u e. v | u =/= (/)} (_ v
13	12	sseli 2036	. . . . . . . 8 |- (z e. {u e. v | u =/= (/)} -> z e. v)
14	12	sseli 2036	. . . . . . . . . 10 |- (w e. {u e. v | u =/= (/)} -> w e. v)
15	14	imim1i 16	. . . . . . . . 9 |- ((w e. v -> ph) -> (w e. {u e. v | u =/= (/)} -> ph))
16	15	r19.20i2 1679	. . . . . . . 8 |- (A.w e. v ph -> A.w e. {u e. v | u =/= (/)}ph)
17	13, 16	imim12i 18	. . . . . . 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	r19.20i2 1679	. . . . . 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 1566	. . . . . . . . 9 |- (u = z -> (u =/= (/) <-> z =/= (/)))
20	19	elrab 1877	. . . . . . . 8 |- (z e. {u e. v | u =/= (/)} <-> (z e. v /\ z =/= (/)))
21	20	pm3.27bi 326	. . . . . . 7 |- (z e. {u e. v | u =/= (/)} -> z =/= (/))
22	21	rgen 1674	. . . . . 6 |- A.z e. {u e. v | u =/= (/)}z =/= (/)
23	18, 22	jctil 292	. . . . 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	biimpr 152	. . . . . . . . 9 |- ((z e. v /\ z =/= (/)) -> z e. {u e. v | u =/= (/)})
25	24	imim1i 16	. . . . . . . 8 |- ((z e. {u e. v | u =/= (/)} -> ps) -> ((z e. v /\ z =/= (/)) -> ps))
26	25	exp3a 375	. . . . . . 7 |- ((z e. {u e. v | u =/= (/)} -> ps) -> (z e. v -> (z =/= (/) -> ps)))
27	26	r19.20i2 1679	. . . . . 6 |- (A.z e. {u e. v | u =/= (/)}ps -> A.z e. v (z =/= (/) -> ps))
28	27	19.22i 1016	. . . . 5 |- (E.yA.z e. {u e. v | u =/= (/)}ps -> E.yA.z e. v (z =/= (/) -> ps))
29	23, 28	imim12i 18	. . . 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	19.20i 968	. . 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 10	. 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		raleq1 1762	. . . . 5 |- (v = x -> (A.w e. v ph <-> A.w e. x ph))
33	32	raleqd 1767	. . . 4 |- (v = x -> (A.z e. v A.w e. v ph <-> A.z e. x A.w e. x ph))
34		raleq1 1762	. . . . 5 |- (v = x -> (A.z e. v (z =/= (/) -> ps) <-> A.z e. x (z =/= (/) -> ps)))
35	34	exbidv 1261	. . . 4 |- (v = x -> (E.yA.z e. v (z =/= (/) -> ps) <-> E.yA.z e. x (z =/= (/) -> ps)))
36	33, 35	imbi12d 624	. . 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 1296	. 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 198	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 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105   =/= wne 1561  A.wral 1621  {crab 1624  (/)c0 2251

This theorem is referenced by:  kmlem13 4701

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rab 1628  df-v 1787  df-in 2022  df-ss 2024

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