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Theorem tfisg 33169
Description: A closed form of tfis 7553. (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 4010 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 dfss3 3906 . . . . . . . . 9 (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
3 nfcv 2958 . . . . . . . . . . . 12 𝑥On
43elrabsf 3767 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
54simprbi 500 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
65ralimi 3131 . . . . . . . . 9 (∀𝑦𝑧 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑧 [𝑦 / 𝑥]𝜑)
72, 6sylbi 220 . . . . . . . 8 (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑧 [𝑦 / 𝑥]𝜑)
8 nfcv 2958 . . . . . . . . . . . 12 𝑥𝑧
9 nfsbc1v 3743 . . . . . . . . . . . 12 𝑥[𝑦 / 𝑥]𝜑
108, 9nfralw 3192 . . . . . . . . . . 11 𝑥𝑦𝑧 [𝑦 / 𝑥]𝜑
11 nfsbc1v 3743 . . . . . . . . . . 11 𝑥[𝑧 / 𝑥]𝜑
1210, 11nfim 1897 . . . . . . . . . 10 𝑥(∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
13 raleq 3361 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑧 [𝑦 / 𝑥]𝜑))
14 sbceq1a 3734 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1513, 14imbi12d 348 . . . . . . . . . 10 (𝑥 = 𝑧 → ((∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ↔ (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
1612, 15rspc 3562 . . . . . . . . 9 (𝑧 ∈ On → (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → (∀𝑦𝑧 [𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)))
1716impcom 411 . . . . . . . 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 488 . . . . . . 7 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
2018, 19jctild 529 . . . . . 6 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑)))
213elrabsf 3767 . . . . . 6 (𝑧 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑧 ∈ On ∧ [𝑧 / 𝑥]𝜑))
2220, 21syl6ibr 255 . . . . 5 ((∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) ∧ 𝑧 ∈ On) → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
2322ralrimiva 3152 . . . 4 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
24 tfi 7552 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
251, 23, 24sylancr 590 . . 3 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → {𝑥 ∈ On ∣ 𝜑} = On)
2625eqcomd 2807 . 2 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → On = {𝑥 ∈ On ∣ 𝜑})
27 rabid2 3337 . 2 (On = {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑥 ∈ On 𝜑)
2826, 27sylib 221 1 (∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥 ∈ On 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  [wsbc 3723  wss 3884  Oncon0 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167
This theorem is referenced by:  soseq  33210
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