Theorem setinds 13844

Description: Principle of _E induction (set induction). If a property passes from all elements of x to x itself, then it holds for all x.

Hypothesis

Ref	Expression
setinds.1	|- (A.y e. x [y / x]ph -> ph)

Assertion

Ref	Expression
setinds	|- ph

Distinct variable groups:   ph,y   x,y

Proof of Theorem setinds

Step	Hyp	Ref	Expression
1		visset 2295	. 2 |- x e. _V
2		setind 5759	. . . . 5 |- (A.z(z C_ {x | ph} -> z e. {x | ph}) -> {x | ph} = _V)
3		df-clab 1872	. . . . . . . 8 |- (y e. {x | ph} <-> [y / x]ph)
4	3	ralbii 2127	. . . . . . 7 |- (A.y e. z y e. {x | ph} <-> A.y e. z [y / x]ph)
5		ax-17 1317	. . . . . . . . . 10 |- (y e. z -> A.x y e. z)
6		hbs1 1722	. . . . . . . . . 10 |- ([y / x]ph -> A.x[y / x]ph)
7	5, 6	hbral 2146	. . . . . . . . 9 |- (A.y e. z [y / x]ph -> A.xA.y e. z [y / x]ph)
8		hbs1 1722	. . . . . . . . 9 |- ([z / x]ph -> A.x[z / x]ph)
9	7, 8	hbim 1354	. . . . . . . 8 |- ((A.y e. z [y / x]ph -> [z / x]ph) -> A.x(A.y e. z [y / x]ph -> [z / x]ph))
10		ax-17 1317	. . . . . . . . . 10 |- (w e. x -> A.y w e. x)
11		ax-17 1317	. . . . . . . . . 10 |- (w e. z -> A.y w e. z)
12	10, 11	raleqf 2263	. . . . . . . . 9 |- (x = z -> (A.y e. x [y / x]ph <-> A.y e. z [y / x]ph))
13		sbequ12 1545	. . . . . . . . 9 |- (x = z -> (ph <-> [z / x]ph))
14	12, 13	imbi12d 688	. . . . . . . 8 |- (x = z -> ((A.y e. x [y / x]ph -> ph) <-> (A.y e. z [y / x]ph -> [z / x]ph)))
15		setinds.1	. . . . . . . 8 |- (A.y e. x [y / x]ph -> ph)
16	9, 14, 15	chvar 1530	. . . . . . 7 |- (A.y e. z [y / x]ph -> [z / x]ph)
17	4, 16	sylbi 216	. . . . . 6 |- (A.y e. z y e. {x | ph} -> [z / x]ph)
18		dfss3 2611	. . . . . 6 |- (z C_ {x | ph} <-> A.y e. z y e. {x | ph})
19		df-clab 1872	. . . . . 6 |- (z e. {x | ph} <-> [z / x]ph)
20	17, 18, 19	3imtr4i 236	. . . . 5 |- (z C_ {x | ph} -> z e. {x | ph})
21	2, 20	mpg 1332	. . . 4 |- {x | ph} = _V
22	21	eqcomi 1888	. . 3 |- _V = {x | ph}
23	22	abeq2i 2001	. 2 |- (x e. _V <-> ph)
24	1, 23	mpbi 206	1 |- ph

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  A.wral 2105  _Vcvv 2292   C_ wss 2593

This theorem is referenced by:  setinds2f 13845

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

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-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140

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