Theorem tfisg 7875

Description: A closed form of tfis 7876. (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 4090	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		dfss3 3984	. . . . . . . . 9 ⊢ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
3		nfcv 2903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥On
4	3	elrabsf 3840	. . . . . . . . . . 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 3081	. . . . . . . . 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 2903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑧
9		nfsbc1v 3811	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
10	8, 9	nfralw 3309	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑
11		nfsbc1v 3811	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
12	10, 11	nfim 1894	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)
13		raleq 3321	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑧 [𝑦 / 𝑥]𝜑))
14		sbceq1a 3802	. . . . . . . . . . 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 3610	. . . . . . . . 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 3840	. . . . . 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 3144	. . . 4 ⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
24		tfi 7874	. . . 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 2741	. 2 ⊢ (∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → On = {𝑥 ∈ On ∣ 𝜑})
27		rabid2 3468	. 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 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  [wsbc 3791   ⊆ wss 3963  Oncon0 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390

This theorem is referenced by:  soseq  8183