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Theorem tfindes 7104
Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction step for successors, and the third is the induction step for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)
Hypotheses
Ref Expression
tfindes.1 [∅ / 𝑥]𝜑
tfindes.2 (𝑥 ∈ On → (𝜑[suc 𝑥 / 𝑥]𝜑))
tfindes.3 (Lim 𝑦 → (∀𝑥𝑦 𝜑[𝑦 / 𝑥]𝜑))
Assertion
Ref Expression
tfindes (𝑥 ∈ On → 𝜑)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem tfindes
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3470 . 2 (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 dfsbcq 3470 . 2 (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑))
3 dfsbcq 3470 . 2 (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
4 sbceq2a 3480 . 2 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
5 tfindes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1883 . . . 4 𝑥 𝑧 ∈ On
7 nfsbc1v 3488 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
8 nfsbc1v 3488 . . . . 5 𝑥[suc 𝑧 / 𝑥]𝜑
97, 8nfim 1865 . . . 4 𝑥([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)
106, 9nfim 1865 . . 3 𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
11 eleq1 2718 . . . 4 (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On))
12 sbceq1a 3479 . . . . 5 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
13 suceq 5828 . . . . . 6 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
1413sbceq1d 3473 . . . . 5 (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
1512, 14imbi12d 333 . . . 4 (𝑥 = 𝑧 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)))
1611, 15imbi12d 333 . . 3 (𝑥 = 𝑧 → ((𝑥 ∈ On → (𝜑[suc 𝑥 / 𝑥]𝜑)) ↔ (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))))
17 tfindes.2 . . 3 (𝑥 ∈ On → (𝜑[suc 𝑥 / 𝑥]𝜑))
1810, 16, 17chvar 2298 . 2 (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
19 cbvralsv 3212 . . . 4 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
20 sbsbc 3472 . . . . 5 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
2120ralbii 3009 . . . 4 (∀𝑧𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
2219, 21bitri 264 . . 3 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
23 tfindes.3 . . 3 (Lim 𝑦 → (∀𝑥𝑦 𝜑[𝑦 / 𝑥]𝜑))
2422, 23syl5bir 233 . 2 (Lim 𝑦 → (∀𝑧𝑦 [𝑧 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
251, 2, 3, 4, 5, 18, 24tfinds 7101 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 1937  wcel 2030  wral 2941  [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:  tfinds2  7105
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