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Theorem tfindes 7010
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 3424 . 2 (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 dfsbcq 3424 . 2 (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑))
3 dfsbcq 3424 . 2 (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
4 sbceq2a 3434 . 2 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
5 tfindes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1845 . . . 4 𝑥 𝑧 ∈ On
7 nfsbc1v 3442 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
8 nfsbc1v 3442 . . . . 5 𝑥[suc 𝑧 / 𝑥]𝜑
97, 8nfim 1827 . . . 4 𝑥([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)
106, 9nfim 1827 . . 3 𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
11 eleq1 2692 . . . 4 (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On))
12 sbceq1a 3433 . . . . 5 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
13 suceq 5752 . . . . . 6 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
1413sbceq1d 3427 . . . . 5 (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
1512, 14imbi12d 334 . . . 4 (𝑥 = 𝑧 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)))
1611, 15imbi12d 334 . . 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 2266 . 2 (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
19 cbvralsv 3175 . . . 4 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
20 sbsbc 3426 . . . . 5 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
2120ralbii 2979 . . . 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 7007 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 1882  wcel 1992  wral 2912  [wsbc 3422  c0 3896  Oncon0 5685  Lim wlim 5686  suc csuc 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691
This theorem is referenced by:  tfinds2  7011
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