Theorem tfindes 7210

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 3590	. 2 ⊢ (𝑦 = ∅ → ([𝑦 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑))
2		dfsbcq 3590	. 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
3		dfsbcq 3590	. 2 ⊢ (𝑦 = suc 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑))
4		sbceq2a 3600	. 2 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
5		tfindes.1	. 2 ⊢ [∅ / 𝑥]𝜑
6		nfv 1995	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ On
7		nfsbc1v 3608	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
8		nfsbc1v 3608	. . . . 5 ⊢ Ⅎ𝑥[suc 𝑧 / 𝑥]𝜑
9	7, 8	nfim 1977	. . . 4 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)
10	6, 9	nfim 1977	. . 3 ⊢ Ⅎ𝑥(𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑))
11		eleq1w 2833	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ On ↔ 𝑧 ∈ On))
12		sbceq1a 3599	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
13		suceq 5934	. . . . . 6 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
14	13	sbceq1d 3593	. . . . 5 ⊢ (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑧 / 𝑥]𝜑))
15	12, 14	imbi12d 333	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑)))
16	11, 15	imbi12d 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))
18	10, 16, 17	chvar 2424	. 2 ⊢ (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑 → [suc 𝑧 / 𝑥]𝜑))
19		cbvralsv 3331	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
20		sbsbc 3592	. . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
21	20	ralbii 3129	. . . 4 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
22	19, 21	bitri 264	. . 3 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
23		tfindes.3	. . 3 ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 𝜑 → [𝑦 / 𝑥]𝜑))
24	22, 23	syl5bir 233	. 2 ⊢ (Lim 𝑦 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
25	1, 2, 3, 4, 5, 18, 24	tfinds 7207	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4  [wsb 2049   ∈ wcel 2145  ∀wral 3061  [wsbc 3588  ∅c0 4064  Oncon0 5867  Lim wlim 5868  suc csuc 5869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7097

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873

This theorem is referenced by:  tfinds2  7211