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Theorem tfinds2 6932
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 4713 . . . 4 ∅ ∈ V
3 tfinds2.1 . . . . 5 (𝑥 = ∅ → (𝜑𝜓))
43imbi2d 328 . . . 4 (𝑥 = ∅ → ((𝜏𝜑) ↔ (𝜏𝜓)))
52, 4sbcie 3436 . . 3 ([∅ / 𝑥](𝜏𝜑) ↔ (𝜏𝜓))
61, 5mpbir 219 . 2 [∅ / 𝑥](𝜏𝜑)
7 vex 3175 . . . . . 6 𝑥 ∈ V
8 tfinds2.5 . . . . . . . 8 (𝑦 ∈ On → (𝜏 → (𝜒𝜃)))
98a2d 29 . . . . . . 7 (𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃)))
109sbcth 3416 . . . . . 6 (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃))))
117, 10ax-mp 5 . . . . 5 [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏𝜒) → (𝜏𝜃)))
12 sbcimg 3443 . . . . . 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 218 . . . 4 ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏𝜒) → (𝜏𝜃)))
15 sbcel1v 3461 . . . 4 ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
16 sbcimg 3443 . . . . 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 278 . . 3 (𝑥 ∈ On → ([𝑥 / 𝑦](𝜏𝜒) → [𝑥 / 𝑦](𝜏𝜃)))
19 tfinds2.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
2019bicomd 211 . . . . . 6 (𝑥 = 𝑦 → (𝜒𝜑))
2120equcoms 1933 . . . . 5 (𝑦 = 𝑥 → (𝜒𝜑))
2221imbi2d 328 . . . 4 (𝑦 = 𝑥 → ((𝜏𝜒) ↔ (𝜏𝜑)))
237, 22sbcie 3436 . . 3 ([𝑥 / 𝑦](𝜏𝜒) ↔ (𝜏𝜑))
24 vex 3175 . . . . . . 7 𝑦 ∈ V
2524sucex 6880 . . . . . 6 suc 𝑦 ∈ V
26 tfinds2.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
2726imbi2d 328 . . . . . 6 (𝑥 = suc 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜃)))
2825, 27sbcie 3436 . . . . 5 ([suc 𝑦 / 𝑥](𝜏𝜑) ↔ (𝜏𝜃))
2928sbcbii 3457 . . . 4 ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏𝜑) ↔ [𝑥 / 𝑦](𝜏𝜃))
30 suceq 5693 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
3130sbcco2 3425 . . . 4 ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏𝜑) ↔ [suc 𝑥 / 𝑥](𝜏𝜑))
3229, 31bitr3i 264 . . 3 ([𝑥 / 𝑦](𝜏𝜃) ↔ [suc 𝑥 / 𝑥](𝜏𝜑))
3318, 23, 323imtr3g 282 . 2 (𝑥 ∈ On → ((𝜏𝜑) → [suc 𝑥 / 𝑥](𝜏𝜑)))
34 sbsbc 3405 . . . 4 ([𝑦 / 𝑥]∀𝑦𝑥 (𝜏𝜒) ↔ [𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒))
3522sbralie 3159 . . . 4 ([𝑦 / 𝑥]∀𝑦𝑥 (𝜏𝜒) ↔ ∀𝑥𝑦 (𝜏𝜑))
3634, 35bitr3i 264 . . 3 ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) ↔ ∀𝑥𝑦 (𝜏𝜑))
37 r19.21v 2942 . . . . . . . 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 230 . . . . . . 7 (Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
4140sbcth 3416 . . . . . 6 (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑))))
4224, 41ax-mp 5 . . . . 5 [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
43 sbcimg 3443 . . . . . 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 218 . . . 4 ([𝑦 / 𝑥]Lim 𝑥[𝑦 / 𝑥](∀𝑦𝑥 (𝜏𝜒) → (𝜏𝜑)))
46 limeq 5638 . . . . 5 (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
4724, 46sbcie 3436 . . . 4 ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
48 sbcimg 3443 . . . . 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 278 . . 3 (Lim 𝑦 → ([𝑦 / 𝑥]𝑦𝑥 (𝜏𝜒) → [𝑦 / 𝑥](𝜏𝜑)))
5136, 50syl5bir 231 . 2 (Lim 𝑦 → (∀𝑥𝑦 (𝜏𝜑) → [𝑦 / 𝑥](𝜏𝜑)))
526, 33, 51tfindes 6931 1 (𝑥 ∈ On → (𝜏𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  [wsb 1866  wcel 1976  wral 2895  Vcvv 3172  [wsbc 3401  c0 3873  Oncon0 5626  Lim wlim 5627  suc csuc 5628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632
This theorem is referenced by:  inar1  9453  grur1a  9497
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