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Theorem tfinds2 7856
Description: Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff 𝜏 is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)
Hypotheses
Ref Expression
tfinds2.1 (𝑥 = ∅ → (𝜑𝜓))
tfinds2.2 (𝑥 = 𝑦 → (𝜑𝜒))
tfinds2.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
tfinds2.4 (𝜏𝜓)
tfinds2.5 (𝑦 ∈ On → (𝜏 → (𝜒𝜃)))
tfinds2.6 (Lim 𝑥 → (𝜏 → (∀𝑦𝑥 𝜒𝜑)))
Assertion
Ref Expression
tfinds2 (𝑥 ∈ On → (𝜏𝜑))
Distinct variable groups:   𝑥,𝑦,𝜏   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)

Proof of Theorem tfinds2
StepHypRef Expression
1 tfinds2.4 . . 3 (𝜏𝜓)
2 0ex 5307 . . . 4 ∅ ∈ V
3 tfinds2.1 . . . . 5 (𝑥 = ∅ → (𝜑𝜓))
43imbi2d 340 . . . 4 (𝑥 = ∅ → ((𝜏𝜑) ↔ (𝜏𝜓)))
52, 4sbcie 3820 . . 3 ([∅ / 𝑥](𝜏𝜑) ↔ (𝜏𝜓))
61, 5mpbir 230 . 2 [∅ / 𝑥](𝜏𝜑)
7 tfinds2.5 . . . . . . . 8 (𝑦 ∈ On → (𝜏 → (𝜒𝜃)))
87a2d 29 . . . . . . 7 (𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃)))
98sbcth 3792 . . . . . 6 (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))))
109elv 3479 . . . . 5 [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃)))
11 sbcimg 3828 . . . . . 6 (𝑥 ∈ V → ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)))))
1211elv 3479 . . . . 5 ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃))))
1310, 12mpbi 229 . . . 4 ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)))
14 sbcel1v 3848 . . . 4 ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
15 sbcimg 3828 . . . . 5 (𝑥 ∈ V → ([𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)) ↔ ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃))))
1615elv 3479 . . . 4 ([𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)) ↔ ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃)))
1713, 14, 163imtr3i 291 . . 3 (𝑥 ∈ On → ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃)))
18 vex 3477 . . . 4 𝑥 ∈ V
19 tfinds2.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
2019bicomd 222 . . . . . 6 (𝑥 = 𝑦 → (𝜒𝜑))
2120equcoms 2022 . . . . 5 (𝑦 = 𝑥 → (𝜒𝜑))
2221imbi2d 340 . . . 4 (𝑦 = 𝑥 → ((𝜏𝜒) ↔ (𝜏𝜑)))
2318, 22sbcie 3820 . . 3 ([𝑥 / 𝑦](𝜏𝜒) ↔ (𝜏𝜑))
24 vex 3477 . . . . . . 7 𝑦 ∈ V
2524sucex 7797 . . . . . 6 suc 𝑦 ∈ V
26 tfinds2.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
2726imbi2d 340 . . . . . 6 (𝑥 = suc 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜃)))
2825, 27sbcie 3820 . . . . 5 ([suc 𝑦 / 𝑥](𝜏𝜑) ↔ (𝜏𝜃))
2928sbcbii 3837 . . . 4 ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏𝜑) ↔ [𝑥 / 𝑦](𝜏𝜃))
30 suceq 6430 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
3130sbcco2 3804 . . . 4 ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏𝜑) ↔ [suc 𝑥 / 𝑥](𝜏𝜑))
3229, 31bitr3i 277 . . 3 ([𝑥 / 𝑦](𝜏𝜃) ↔ [suc 𝑥 / 𝑥](𝜏𝜑))
3317, 23, 323imtr3g 295 . 2 (𝑥 ∈ On → ((𝜏𝜑) → [suc 𝑥 / 𝑥](𝜏𝜑)))
3419imbi2d 340 . . . . 5 (𝑥 = 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜒)))
3534sbralie 3353 . . . 4 (∀𝑥𝑦 (𝜏𝜑) ↔ [𝑦 / 𝑥]∀𝑦𝑥 (𝜏𝜒))
36 sbsbc 3781 . . . 4 ([𝑦 / 𝑥]∀𝑦𝑥 (𝜏𝜒) ↔ [𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒))
3735, 36bitr2i 276 . . 3 ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) ↔ ∀𝑥𝑦 (𝜏𝜑))
38 r19.21v 3178 . . . . . . . 8 (∀𝑦𝑥 (𝜏𝜒) ↔ (𝜏 → ∀𝑦𝑥 𝜒))
39 tfinds2.6 . . . . . . . . 9 (Lim 𝑥 → (𝜏 → (∀𝑦𝑥 𝜒𝜑)))
4039a2d 29 . . . . . . . 8 (Lim 𝑥 → ((𝜏 → ∀𝑦𝑥 𝜒) → (𝜏𝜑)))
4138, 40biimtrid 241 . . . . . . 7 (Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
4241sbcth 3792 . . . . . 6 (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))))
4342elv 3479 . . . . 5 [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
44 sbcimg 3828 . . . . . 6 (𝑦 ∈ V → ([𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))) ↔ ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))))
4544elv 3479 . . . . 5 ([𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))) ↔ ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))))
4643, 45mpbi 229 . . . 4 ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
47 limeq 6376 . . . . 5 (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
4824, 47sbcie 3820 . . . 4 ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
49 sbcimg 3828 . . . . 5 (𝑦 ∈ V → ([𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)) ↔ ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑))))
5049elv 3479 . . . 4 ([𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)) ↔ ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑)))
5146, 48, 503imtr3i 291 . . 3 (Lim 𝑦 → ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑)))
5237, 51biimtrrid 242 . 2 (Lim 𝑦 → (∀𝑥𝑦 (𝜏𝜑) → [𝑦 / 𝑥](𝜏𝜑)))
536, 33, 52tfindes 7855 1 (𝑥 ∈ On → (𝜏𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  [wsb 2066  wcel 2105  wral 3060  Vcvv 3473  [wsbc 3777  c0 4322  Oncon0 6364  Lim wlim 6365  suc csuc 6366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370
This theorem is referenced by:  inar1  10773  grur1a  10817
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