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Theorem tfinds2 7105
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 4823 . . . 4 ∅ ∈ V
3 tfinds2.1 . . . . 5 (𝑥 = ∅ → (𝜑𝜓))
43imbi2d 329 . . . 4 (𝑥 = ∅ → ((𝜏𝜑) ↔ (𝜏𝜓)))
52, 4sbcie 3503 . . 3 ([∅ / 𝑥](𝜏𝜑) ↔ (𝜏𝜓))
61, 5mpbir 221 . 2 [∅ / 𝑥](𝜏𝜑)
7 vex 3234 . . . . . 6 𝑥 ∈ V
8 tfinds2.5 . . . . . . . 8 (𝑦 ∈ On → (𝜏 → (𝜒𝜃)))
98a2d 29 . . . . . . 7 (𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃)))
109sbcth 3483 . . . . . 6 (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))))
117, 10ax-mp 5 . . . . 5 [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃)))
12 sbcimg 3510 . . . . . 6 (𝑥 ∈ V → ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)))))
137, 12ax-mp 5 . . . . 5 ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃))))
1411, 13mpbi 220 . . . 4 ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)))
15 sbcel1v 3528 . . . 4 ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
16 sbcimg 3510 . . . . 5 (𝑥 ∈ V → ([𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)) ↔ ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃))))
177, 16ax-mp 5 . . . 4 ([𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)) ↔ ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃)))
1814, 15, 173imtr3i 280 . . 3 (𝑥 ∈ On → ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃)))
19 tfinds2.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
2019bicomd 213 . . . . . 6 (𝑥 = 𝑦 → (𝜒𝜑))
2120equcoms 1993 . . . . 5 (𝑦 = 𝑥 → (𝜒𝜑))
2221imbi2d 329 . . . 4 (𝑦 = 𝑥 → ((𝜏𝜒) ↔ (𝜏𝜑)))
237, 22sbcie 3503 . . 3 ([𝑥 / 𝑦](𝜏𝜒) ↔ (𝜏𝜑))
24 vex 3234 . . . . . . 7 𝑦 ∈ V
2524sucex 7053 . . . . . 6 suc 𝑦 ∈ V
26 tfinds2.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
2726imbi2d 329 . . . . . 6 (𝑥 = suc 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜃)))
2825, 27sbcie 3503 . . . . 5 ([suc 𝑦 / 𝑥](𝜏𝜑) ↔ (𝜏𝜃))
2928sbcbii 3524 . . . 4 ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏𝜑) ↔ [𝑥 / 𝑦](𝜏𝜃))
30 suceq 5828 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
3130sbcco2 3492 . . . 4 ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏𝜑) ↔ [suc 𝑥 / 𝑥](𝜏𝜑))
3229, 31bitr3i 266 . . 3 ([𝑥 / 𝑦](𝜏𝜃) ↔ [suc 𝑥 / 𝑥](𝜏𝜑))
3318, 23, 323imtr3g 284 . 2 (𝑥 ∈ On → ((𝜏𝜑) → [suc 𝑥 / 𝑥](𝜏𝜑)))
34 sbsbc 3472 . . . 4 ([𝑦 / 𝑥]∀𝑦𝑥 (𝜏𝜒) ↔ [𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒))
3522sbralie 3214 . . . 4 ([𝑦 / 𝑥]∀𝑦𝑥 (𝜏𝜒) ↔ ∀𝑥𝑦 (𝜏𝜑))
3634, 35bitr3i 266 . . 3 ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) ↔ ∀𝑥𝑦 (𝜏𝜑))
37 r19.21v 2989 . . . . . . . 8 (∀𝑦𝑥 (𝜏𝜒) ↔ (𝜏 → ∀𝑦𝑥 𝜒))
38 tfinds2.6 . . . . . . . . 9 (Lim 𝑥 → (𝜏 → (∀𝑦𝑥 𝜒𝜑)))
3938a2d 29 . . . . . . . 8 (Lim 𝑥 → ((𝜏 → ∀𝑦𝑥 𝜒) → (𝜏𝜑)))
4037, 39syl5bi 232 . . . . . . 7 (Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
4140sbcth 3483 . . . . . 6 (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))))
4224, 41ax-mp 5 . . . . 5 [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
43 sbcimg 3510 . . . . . 6 (𝑦 ∈ V → ([𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))) ↔ ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))))
4424, 43ax-mp 5 . . . . 5 ([𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))) ↔ ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))))
4542, 44mpbi 220 . . . 4 ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
46 limeq 5773 . . . . 5 (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
4724, 46sbcie 3503 . . . 4 ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
48 sbcimg 3510 . . . . 5 (𝑦 ∈ V → ([𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)) ↔ ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑))))
4924, 48ax-mp 5 . . . 4 ([𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)) ↔ ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑)))
5045, 47, 493imtr3i 280 . . 3 (Lim 𝑦 → ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑)))
5136, 50syl5bir 233 . 2 (Lim 𝑦 → (∀𝑥𝑦 (𝜏𝜑) → [𝑦 / 𝑥](𝜏𝜑)))
526, 33, 51tfindes 7104 1 (𝑥 ∈ On → (𝜏𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  [wsb 1937  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  c0 3948  Oncon0 5761  Lim wlim 5762  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767
This theorem is referenced by:  inar1  9635  grur1a  9679
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