Theorem tfisg 13912

Description: A closed form of tfis 3938.

Assertion

Ref	Expression
tfisg	|- (A.x e. On (A.y e. x [y / x]ph -> ph) -> A.x e. On ph)

Distinct variable groups:   ph,y   x,y

Proof of Theorem tfisg

Step	Hyp	Ref	Expression
1		tfi 3937	. . . 4 |- (({x e. On | ph} C_ On /\ A.z e. On (z C_ {x e. On | ph} -> z e. {x e. On | ph})) -> {x e. On | ph} = On)
2		ssrab2 2692	. . . 4 |- {x e. On | ph} C_ On
3		ax-17 1317	. . . . . . . . . . . 12 |- (y e. z -> A.x y e. z)
4		hbs1 1722	. . . . . . . . . . . 12 |- ([y / x]ph -> A.x[y / x]ph)
5	3, 4	hbral 2146	. . . . . . . . . . 11 |- (A.y e. z [y / x]ph -> A.xA.y e. z [y / x]ph)
6		hbs1 1722	. . . . . . . . . . 11 |- ([z / x]ph -> A.x[z / x]ph)
7	5, 6	hbim 1354	. . . . . . . . . 10 |- ((A.y e. z [y / x]ph -> [z / x]ph) -> A.x(A.y e. z [y / x]ph -> [z / x]ph))
8		raleq 2266	. . . . . . . . . . 11 |- (x = z -> (A.y e. x [y / x]ph <-> A.y e. z [y / x]ph))
9		sbequ12 1545	. . . . . . . . . . 11 |- (x = z -> (ph <-> [z / x]ph))
10	8, 9	imbi12d 688	. . . . . . . . . 10 |- (x = z -> ((A.y e. x [y / x]ph -> ph) <-> (A.y e. z [y / x]ph -> [z / x]ph)))
11	7, 10	rcla4 2373	. . . . . . . . 9 |- (z e. On -> (A.x e. On (A.y e. x [y / x]ph -> ph) -> (A.y e. z [y / x]ph -> [z / x]ph)))
12	11	impcom 378	. . . . . . . 8 |- ((A.x e. On (A.y e. x [y / x]ph -> ph) /\ z e. On) -> (A.y e. z [y / x]ph -> [z / x]ph))
13		dfss3 2611	. . . . . . . . 9 |- (z C_ {x e. On | ph} <-> A.y e. z y e. {x e. On | ph})
14		ax-17 1317	. . . . . . . . . . . 12 |- (z e. On -> A.x z e. On)
15	14	elrabsf 2486	. . . . . . . . . . 11 |- (y e. {x e. On | ph} <-> (y e. On /\ [y / x]ph))
16	15	simprbi 353	. . . . . . . . . 10 |- (y e. {x e. On | ph} -> [y / x]ph)
17	16	ralimi 2168	. . . . . . . . 9 |- (A.y e. z y e. {x e. On | ph} -> A.y e. z [y / x]ph)
18	13, 17	sylbi 216	. . . . . . . 8 |- (z C_ {x e. On | ph} -> A.y e. z [y / x]ph)
19	12, 18	syl5 20	. . . . . . 7 |- ((A.x e. On (A.y e. x [y / x]ph -> ph) /\ z e. On) -> (z C_ {x e. On | ph} -> [z / x]ph))
20		simpr 350	. . . . . . 7 |- ((A.x e. On (A.y e. x [y / x]ph -> ph) /\ z e. On) -> z e. On)
21	19, 20	jctild 662	. . . . . 6 |- ((A.x e. On (A.y e. x [y / x]ph -> ph) /\ z e. On) -> (z C_ {x e. On | ph} -> (z e. On /\ [z / x]ph)))
22	14	elrabsf 2486	. . . . . 6 |- (z e. {x e. On | ph} <-> (z e. On /\ [z / x]ph))
23	21, 22	syl6ibr 230	. . . . 5 |- ((A.x e. On (A.y e. x [y / x]ph -> ph) /\ z e. On) -> (z C_ {x e. On | ph} -> z e. {x e. On | ph}))
24	23	r19.21aiva 2176	. . . 4 |- (A.x e. On (A.y e. x [y / x]ph -> ph) -> A.z e. On (z C_ {x e. On | ph} -> z e. {x e. On | ph}))
25	1, 2, 24	sylancr 526	. . 3 |- (A.x e. On (A.y e. x [y / x]ph -> ph) -> {x e. On | ph} = On)
26	25	eqcomd 1889	. 2 |- (A.x e. On (A.y e. x [y / x]ph -> ph) -> On = {x e. On | ph})
27		rabid2 2254	. 2 |- (On = {x e. On | ph} <-> A.x e. On ph)
28	26, 27	sylib 215	1 |- (A.x e. On (A.y e. x [y / x]ph -> ph) -> A.x e. On ph)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  {crab 2108   C_ wss 2593  Oncon0 3657

This theorem is referenced by:  soseq 13955

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-13 1311  ax-14 1312  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  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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