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Theorem tfisg 30802
Description: A closed form of tfis 6822. (Contributed by Scott Fenton, 8-Jun-2011.)
Assertion
Ref Expression
tfisg (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀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.
StepHypRef Expression
1 ssrab2 3554 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 dfss3 3462 . . . . . . . . 9 (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
3 nfcv 2655 . . . . . . . . . . . 12 𝑥On
43elrabsf 3345 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
54simprbi 478 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
65ralimi 2840 . . . . . . . . 9 (∀𝑦𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑧 [𝑦 / 𝑥]𝜑)
72, 6sylbi 205 . . . . . . . 8 (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑧 [𝑦 / 𝑥]𝜑)
8 nfcv 2655 . . . . . . . . . . . 12 𝑥𝑧
9 nfsbc1v 3326 . . . . . . . . . . . 12 𝑥[𝑦 / 𝑥]𝜑
108, 9nfral 2833 . . . . . . . . . . 11 𝑥𝑦𝑧 [𝑦 / 𝑥]𝜑
11 nfsbc1v 3326 . . . . . . . . . . 11 𝑥[𝑧 / 𝑥]𝜑
1210, 11nfim 2051 . . . . . . . . . 10 𝑥(∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
13 raleq 3019 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 [𝑦 / 𝑥]𝜑))
14 sbceq1a 3317 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1513, 14imbi12d 332 . . . . . . . . . 10 (𝑥 = 𝑧 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ↔ (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
1612, 15rspc 3180 . . . . . . . . 9 (𝑧 ∈ On → (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
1716impcom 444 . . . . . . . 8 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑))
187, 17syl5 33 . . . . . . 7 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → [𝑧 / 𝑥]𝜑))
19 simpr 475 . . . . . . 7 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
2018, 19jctild 563 . . . . . 6 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑)))
213elrabsf 3345 . . . . . 6 (𝑧 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑))
2220, 21syl6ibr 240 . . . . 5 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
2322ralrimiva 2853 . . . 4 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
24 tfi 6821 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
251, 23, 24sylancr 693 . . 3 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → {𝑥 ∈ On ∣ 𝜑} = On)
2625eqcomd 2520 . 2 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → On = {𝑥 ∈ On ∣ 𝜑})
27 rabid2 3000 . 2 (On = {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑥 ∈ On 𝜑)
2826, 27sylib 206 1 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥 ∈ On 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  wral 2800  {crab 2804  [wsbc 3306  wss 3444  Oncon0 5530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-tr 4579  df-eprel 4843  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-ord 5533  df-on 5534
This theorem is referenced by:  soseq  30837
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