Theorem setinds 35742

Description: Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by Scott Fenton, 10-Mar-2011.)

Hypothesis

Ref	Expression
setinds.1	⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑)

Assertion

Ref	Expression
setinds	⊢ 𝜑

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem setinds

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3492	. 2 ⊢ 𝑥 ∈ V
2		setind 9803	. . . . 5 ⊢ (∀𝑧(𝑧 ⊆ {𝑥 ∣ 𝜑} → 𝑧 ∈ {𝑥 ∣ 𝜑}) → {𝑥 ∣ 𝜑} = V)
3		dfss3 3997	. . . . . . 7 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑})
4		df-sbc 3805	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑})
5	4	ralbii 3099	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑})
6		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑧
7		nfsbc1v 3824	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
8	6, 7	nfralw 3317	. . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑
9		nfsbc1v 3824	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
10	8, 9	nfim 1895	. . . . . . . . 9 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)
11		raleq 3331	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑))
12		sbceq1a 3815	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
13	11, 12	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ↔ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)))
14		setinds.1	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑)
15	10, 13, 14	chvarfv 2241	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)
16	5, 15	sylbir 235	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∣ 𝜑} → [𝑧 / 𝑥]𝜑)
17	3, 16	sylbi 217	. . . . . 6 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} → [𝑧 / 𝑥]𝜑)
18		df-sbc 3805	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ {𝑥 ∣ 𝜑})
19	17, 18	sylib 218	. . . . 5 ⊢ (𝑧 ⊆ {𝑥 ∣ 𝜑} → 𝑧 ∈ {𝑥 ∣ 𝜑})
20	2, 19	mpg 1795	. . . 4 ⊢ {𝑥 ∣ 𝜑} = V
21	20	eqcomi 2749	. . 3 ⊢ V = {𝑥 ∣ 𝜑}
22	21	eqabri 2888	. 2 ⊢ (𝑥 ∈ V ↔ 𝜑)
23	1, 22	mpbi 230	1 ⊢ 𝜑

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  Vcvv 3488  [wsbc 3804   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466

This theorem is referenced by:  setinds2f  35743