Theorem tfisg 7768

Description: A closed form of tfis 7769. (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 4025	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		dfss3 3920	. . . . . . . . 9 ⊢ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
3		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥On
4	3	elrabsf 3775	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
5	4	simprbi 497	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
6	5	ralimi 3082	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑)
7	2, 6	sylbi 216	. . . . . . . 8 ⊢ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑)
8		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑧
9		nfsbc1v 3747	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
10	8, 9	nfralw 3290	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑
11		nfsbc1v 3747	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
12	10, 11	nfim 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)
13		raleq 3305	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑))
14		sbceq1a 3738	. . . . . . . . . . 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 3558	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → (∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)))
17	16	impcom 408	. . . . . . . 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 485	. . . . . . 7 ⊢ ((∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
20	18, 19	jctild 526	. . . . . 6 ⊢ ((∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑)))
21	3	elrabsf 3775	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑))
22	20, 21	syl6ibr 251	. . . . 5 ⊢ ((∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
23	22	ralrimiva 3139	. . . 4 ⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
24		tfi 7767	. . . 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 3432	. 2 ⊢ (On = {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑥 ∈ On 𝜑)
28	26, 27	sylib 217	1 ⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥 ∈ On 𝜑)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061  {crab 3403  [wsbc 3727   ⊆ wss 3898  Oncon0 6302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-tr 5210  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-ord 6305  df-on 6306

This theorem is referenced by:  soseq  8046