Theorem tfisg 7854

Description: A closed form of tfis 7855. (Contributed by Scott Fenton, 8-Jun-2011.)

Assertion

Ref	Expression
tfisg	⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥 ∈ On 𝜑)

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem tfisg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4060	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		dfss3 3952	. . . . . . . . 9 ⊢ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
3		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥On
4	3	elrabsf 3816	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
5	4	simprbi 496	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
6	5	ralimi 3074	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑)
7	2, 6	sylbi 217	. . . . . . . 8 ⊢ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑)
8		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑧
9		nfsbc1v 3790	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
10	8, 9	nfralw 3295	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑
11		nfsbc1v 3790	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
12	10, 11	nfim 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)
13		raleq 3306	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑))
14		sbceq1a 3781	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
15	13, 14	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ↔ (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)))
16	12, 15	rspc 3594	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)))
17	16	impcom 407	. . . . . . . 8 ⊢ ((∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ∧ 𝑧 ∈ On) → (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
18	7, 17	syl5 34	. . . . . . 7 ⊢ ((∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → [𝑧 / 𝑥]𝜑))
19		simpr 484	. . . . . . 7 ⊢ ((∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
20	18, 19	jctild 525	. . . . . 6 ⊢ ((∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑)))
21	3	elrabsf 3816	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑))
22	20, 21	imbitrrdi 252	. . . . 5 ⊢ ((∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
23	22	ralrimiva 3133	. . . 4 ⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
24		tfi 7853	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
25	1, 23, 24	sylancr 587	. . 3 ⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → {𝑥 ∈ On ∣ 𝜑} = On)
26	25	eqcomd 2742	. 2 ⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → On = {𝑥 ∈ On ∣ 𝜑})
27		rabid2 3454	. 2 ⊢ (On = {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑥 ∈ On 𝜑)
28	26, 27	sylib 218	1 ⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥 ∈ On 𝜑)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420  [wsbc 3770   ⊆ wss 3931  Oncon0 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361

This theorem is referenced by:  soseq  8163