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Theorem tfindes 7342
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 3654 . 2 (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 dfsbcq 3654 . 2 (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑))
3 dfsbcq 3654 . 2 (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
4 sbceq2a 3664 . 2 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
5 tfindes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1957 . . . 4 𝑥 𝑧 ∈ On
7 nfsbc1v 3672 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
8 nfsbc1v 3672 . . . . 5 𝑥[suc 𝑧 / 𝑥]𝜑
97, 8nfim 1943 . . . 4 𝑥([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)
106, 9nfim 1943 . . 3 𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
11 eleq1w 2842 . . . 4 (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On))
12 sbceq1a 3663 . . . . 5 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
13 suceq 6043 . . . . . 6 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
1413sbceq1d 3657 . . . . 5 (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
1512, 14imbi12d 336 . . . 4 (𝑥 = 𝑧 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)))
1611, 15imbi12d 336 . . 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 2360 . 2 (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
19 cbvralsv 3378 . . . 4 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
20 sbsbc 3656 . . . . 5 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
2120ralbii 3162 . . . 4 (∀𝑧𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
2219, 21bitri 267 . . 3 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
23 tfindes.3 . . 3 (Lim 𝑦 → (∀𝑥𝑦 𝜑[𝑦 / 𝑥]𝜑))
2422, 23syl5bir 235 . 2 (Lim 𝑦 → (∀𝑧𝑦 [𝑧 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
251, 2, 3, 4, 5, 18, 24tfinds 7339 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2011  wcel 2107  wral 3090  [wsbc 3652  c0 4141  Oncon0 5978  Lim wlim 5979  suc csuc 5980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-tr 4990  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984
This theorem is referenced by:  tfinds2  7343
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