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Theorem tfinds2 7563
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 5187 . . . 4 ∅ ∈ V
3 tfinds2.1 . . . . 5 (𝑥 = ∅ → (𝜑𝜓))
43imbi2d 344 . . . 4 (𝑥 = ∅ → ((𝜏𝜑) ↔ (𝜏𝜓)))
52, 4sbcie 3787 . . 3 ([∅ / 𝑥](𝜏𝜑) ↔ (𝜏𝜓))
61, 5mpbir 234 . 2 [∅ / 𝑥](𝜏𝜑)
7 tfinds2.5 . . . . . . . 8 (𝑦 ∈ On → (𝜏 → (𝜒𝜃)))
87a2d 29 . . . . . . 7 (𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃)))
98sbcth 3762 . . . . . 6 (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))))
109elv 3474 . . . . 5 [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃)))
11 sbcimg 3794 . . . . . 6 (𝑥 ∈ V → ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)))))
1211elv 3474 . . . . 5 ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃))))
1310, 12mpbi 233 . . . 4 ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)))
14 sbcel1v 3813 . . . 4 ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
15 sbcimg 3794 . . . . 5 (𝑥 ∈ V → ([𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)) ↔ ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃))))
1615elv 3474 . . . 4 ([𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)) ↔ ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃)))
1713, 14, 163imtr3i 294 . . 3 (𝑥 ∈ On → ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃)))
18 vex 3472 . . . 4 𝑥 ∈ V
19 tfinds2.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
2019bicomd 226 . . . . . 6 (𝑥 = 𝑦 → (𝜒𝜑))
2120equcoms 2027 . . . . 5 (𝑦 = 𝑥 → (𝜒𝜑))
2221imbi2d 344 . . . 4 (𝑦 = 𝑥 → ((𝜏𝜒) ↔ (𝜏𝜑)))
2318, 22sbcie 3787 . . 3 ([𝑥 / 𝑦](𝜏𝜒) ↔ (𝜏𝜑))
24 vex 3472 . . . . . . 7 𝑦 ∈ V
2524sucex 7511 . . . . . 6 suc 𝑦 ∈ V
26 tfinds2.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
2726imbi2d 344 . . . . . 6 (𝑥 = suc 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜃)))
2825, 27sbcie 3787 . . . . 5 ([suc 𝑦 / 𝑥](𝜏𝜑) ↔ (𝜏𝜃))
2928sbcbii 3803 . . . 4 ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏𝜑) ↔ [𝑥 / 𝑦](𝜏𝜃))
30 suceq 6234 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
3130sbcco2 3774 . . . 4 ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏𝜑) ↔ [suc 𝑥 / 𝑥](𝜏𝜑))
3229, 31bitr3i 280 . . 3 ([𝑥 / 𝑦](𝜏𝜃) ↔ [suc 𝑥 / 𝑥](𝜏𝜑))
3317, 23, 323imtr3g 298 . 2 (𝑥 ∈ On → ((𝜏𝜑) → [suc 𝑥 / 𝑥](𝜏𝜑)))
3419imbi2d 344 . . . . 5 (𝑥 = 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜒)))
3534sbralie 3446 . . . 4 (∀𝑥𝑦 (𝜏𝜑) ↔ [𝑦 / 𝑥]∀𝑦𝑥 (𝜏𝜒))
36 sbsbc 3751 . . . 4 ([𝑦 / 𝑥]∀𝑦𝑥 (𝜏𝜒) ↔ [𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒))
3735, 36bitr2i 279 . . 3 ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) ↔ ∀𝑥𝑦 (𝜏𝜑))
38 r19.21v 3167 . . . . . . . 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, 40syl5bi 245 . . . . . . 7 (Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
4241sbcth 3762 . . . . . 6 (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))))
4342elv 3474 . . . . 5 [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
44 sbcimg 3794 . . . . . 6 (𝑦 ∈ V → ([𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))) ↔ ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))))
4544elv 3474 . . . . 5 ([𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))) ↔ ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))))
4643, 45mpbi 233 . . . 4 ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
47 limeq 6181 . . . . 5 (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
4824, 47sbcie 3787 . . . 4 ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
49 sbcimg 3794 . . . . 5 (𝑦 ∈ V → ([𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)) ↔ ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑))))
5049elv 3474 . . . 4 ([𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)) ↔ ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑)))
5146, 48, 503imtr3i 294 . . 3 (Lim 𝑦 → ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑)))
5237, 51syl5bir 246 . 2 (Lim 𝑦 → (∀𝑥𝑦 (𝜏𝜑) → [𝑦 / 𝑥](𝜏𝜑)))
536, 33, 52tfindes 7562 1 (𝑥 ∈ On → (𝜏𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  [wsb 2069  wcel 2114  wral 3130  Vcvv 3469  [wsbc 3747  c0 4265  Oncon0 6169  Lim wlim 6170  suc csuc 6171
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175
This theorem is referenced by:  inar1  10186  grur1a  10230
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