Theorem tfindes 7803

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 → (∀𝑥 ∈ 𝑦 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.

Step	Hyp	Ref	Expression
1		dfsbcq 3725	. 2 ⊢ (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑))
2		dfsbcq 3725	. 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
3		dfsbcq 3725	. 2 ⊢ (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑))
4		sbceq2a 3735	. 2 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
5		tfindes.1	. 2 ⊢ [∅ / 𝑥]𝜑
6		nfv 1921	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ On
7		nfsbc1v 3743	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
8		nfsbc1v 3743	. . . . 5 ⊢ Ⅎ𝑥[suc 𝑧 / 𝑥]𝜑
9	7, 8	nfim 1903	. . . 4 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)
10	6, 9	nfim 1903	. . 3 ⊢ Ⅎ𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑))
11		eleq1w 2822	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On))
12		sbceq1a 3734	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
13		suceq 6378	. . . . . 6 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
14	13	sbceq1d 3728	. . . . 5 ⊢ (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑))
15	12, 14	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)))
16	11, 15	imbi12d 345	. . 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))
18	10, 16, 17	chvarfv 2252	. 2 ⊢ (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑))
19		cbvralsvw 3290	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
20		sbsbc 3727	. . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
21	20	ralbii 3085	. . . 4 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
22	19, 21	bitri 276	. . 3 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
23		tfindes.3	. . 3 ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑))
24	22, 23	biimtrrid 244	. 2 ⊢ (Lim 𝑦 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
25	1, 2, 3, 4, 5, 18, 24	tfinds 7800	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4  [wsb 2073   ∈ wcel 2119  ∀wral 3053  [wsbc 3723  ∅c0 4261  Oncon0 6310  Lim wlim 6311  suc csuc 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316

This theorem is referenced by:  tfinds2  7804  rdgssun  37740