MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfindes Structured version   Visualization version   GIF version

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 → (∀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 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]𝜑𝜑))
5 tfindes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1995 . . . 4 𝑥 𝑧 ∈ On
7 nfsbc1v 3608 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
8 nfsbc1v 3608 . . . . 5 𝑥[suc 𝑧 / 𝑥]𝜑
97, 8nfim 1977 . . . 4 𝑥([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)
106, 9nfim 1977 . . 3 𝑥(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)
1413sbceq1d 3593 . . . . 5 (𝑥 = 𝑧 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
1512, 14imbi12d 333 . . . 4 (𝑥 = 𝑧 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑)))
1611, 15imbi12d 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))
1810, 16, 17chvar 2424 . 2 (𝑧 ∈ On → ([𝑧 / 𝑥]𝜑[suc 𝑧 / 𝑥]𝜑))
19 cbvralsv 3331 . . . 4 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
20 sbsbc 3592 . . . . 5 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
2120ralbii 3129 . . . 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 7207 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator