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Theorem tfindes 7884
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 3793 . 2 (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 dfsbcq 3793 . 2 (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑))
3 dfsbcq 3793 . 2 (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
4 sbceq2a 3803 . 2 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
5 tfindes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1912 . . . 4 𝑥 𝑧 ∈ On
7 nfsbc1v 3811 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
8 nfsbc1v 3811 . . . . 5 𝑥[suc 𝑧 / 𝑥]𝜑
97, 8nfim 1894 . . . 4 𝑥([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)
106, 9nfim 1894 . . 3 𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
11 eleq1w 2822 . . . 4 (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On))
12 sbceq1a 3802 . . . . 5 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
13 suceq 6452 . . . . . 6 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
1413sbceq1d 3796 . . . . 5 (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
1512, 14imbi12d 344 . . . 4 (𝑥 = 𝑧 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)))
1611, 15imbi12d 344 . . 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, 17chvarfv 2238 . 2 (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
19 cbvralsvw 3315 . . . 4 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
20 sbsbc 3795 . . . . 5 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
2120ralbii 3091 . . . 4 (∀𝑧𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
2219, 21bitri 275 . . 3 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
23 tfindes.3 . . 3 (Lim 𝑦 → (∀𝑥𝑦 𝜑[𝑦 / 𝑥]𝜑))
2422, 23biimtrrid 243 . 2 (Lim 𝑦 → (∀𝑧𝑦 [𝑧 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
251, 2, 3, 4, 5, 18, 24tfinds 7881 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2062  wcel 2106  wral 3059  [wsbc 3791  c0 4339  Oncon0 6386  Lim wlim 6387  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392
This theorem is referenced by:  tfinds2  7885  rdgssun  37361
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