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Theorem tfisg 7795
Description: A closed form of tfis 7796. (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 4042 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 dfss3 3937 . . . . . . . . 9 (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
3 nfcv 2908 . . . . . . . . . . . 12 𝑥On
43elrabsf 3792 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
54simprbi 498 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
65ralimi 3087 . . . . . . . . 9 (∀𝑦𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑧 [𝑦 / 𝑥]𝜑)
72, 6sylbi 216 . . . . . . . 8 (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑧 [𝑦 / 𝑥]𝜑)
8 nfcv 2908 . . . . . . . . . . . 12 𝑥𝑧
9 nfsbc1v 3764 . . . . . . . . . . . 12 𝑥[𝑦 / 𝑥]𝜑
108, 9nfralw 3297 . . . . . . . . . . 11 𝑥𝑦𝑧 [𝑦 / 𝑥]𝜑
11 nfsbc1v 3764 . . . . . . . . . . 11 𝑥[𝑧 / 𝑥]𝜑
1210, 11nfim 1900 . . . . . . . . . 10 𝑥(∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
13 raleq 3312 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 [𝑦 / 𝑥]𝜑))
14 sbceq1a 3755 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1513, 14imbi12d 345 . . . . . . . . . 10 (𝑥 = 𝑧 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ↔ (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
1612, 15rspc 3572 . . . . . . . . 9 (𝑧 ∈ On → (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
1716impcom 409 . . . . . . . 8 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑))
187, 17syl5 34 . . . . . . 7 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → [𝑧 / 𝑥]𝜑))
19 simpr 486 . . . . . . 7 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
2018, 19jctild 527 . . . . . 6 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑)))
213elrabsf 3792 . . . . . 6 (𝑧 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑))
2220, 21syl6ibr 252 . . . . 5 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
2322ralrimiva 3144 . . . 4 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
24 tfi 7794 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
251, 23, 24sylancr 588 . . 3 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → {𝑥 ∈ On ∣ 𝜑} = On)
2625eqcomd 2743 . 2 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → On = {𝑥 ∈ On ∣ 𝜑})
27 rabid2 3439 . 2 (On = {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑥 ∈ On 𝜑)
2826, 27sylib 217 1 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥 ∈ On 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  {crab 3410  [wsbc 3744  wss 3915  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  soseq  8096
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