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Theorem tfisg 7876
Description: A closed form of tfis 7877. (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 4079 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 dfss3 3971 . . . . . . . . 9 (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
3 nfcv 2904 . . . . . . . . . . . 12 𝑥On
43elrabsf 3833 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
54simprbi 496 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
65ralimi 3082 . . . . . . . . 9 (∀𝑦𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑧 [𝑦 / 𝑥]𝜑)
72, 6sylbi 217 . . . . . . . 8 (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑧 [𝑦 / 𝑥]𝜑)
8 nfcv 2904 . . . . . . . . . . . 12 𝑥𝑧
9 nfsbc1v 3807 . . . . . . . . . . . 12 𝑥[𝑦 / 𝑥]𝜑
108, 9nfralw 3310 . . . . . . . . . . 11 𝑥𝑦𝑧 [𝑦 / 𝑥]𝜑
11 nfsbc1v 3807 . . . . . . . . . . 11 𝑥[𝑧 / 𝑥]𝜑
1210, 11nfim 1895 . . . . . . . . . 10 𝑥(∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
13 raleq 3322 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 [𝑦 / 𝑥]𝜑))
14 sbceq1a 3798 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1513, 14imbi12d 344 . . . . . . . . . 10 (𝑥 = 𝑧 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ↔ (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
1612, 15rspc 3609 . . . . . . . . 9 (𝑧 ∈ On → (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
1716impcom 407 . . . . . . . 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 484 . . . . . . 7 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
2018, 19jctild 525 . . . . . 6 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑)))
213elrabsf 3833 . . . . . 6 (𝑧 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑))
2220, 21imbitrrdi 252 . . . . 5 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
2322ralrimiva 3145 . . . 4 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
24 tfi 7875 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
251, 23, 24sylancr 587 . . 3 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → {𝑥 ∈ On ∣ 𝜑} = On)
2625eqcomd 2742 . 2 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → On = {𝑥 ∈ On ∣ 𝜑})
27 rabid2 3469 . 2 (On = {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑥 ∈ On 𝜑)
2826, 27sylib 218 1 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥 ∈ On 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  [wsbc 3787  wss 3950  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by:  soseq  8185
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