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Theorem on2ind 8616
Description: Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.)
Hypotheses
Ref Expression
on2ind.1 (𝑎 = 𝑐 → (𝜑𝜓))
on2ind.2 (𝑏 = 𝑑 → (𝜓𝜒))
on2ind.3 (𝑎 = 𝑐 → (𝜃𝜒))
on2ind.4 (𝑎 = 𝑋 → (𝜑𝜏))
on2ind.5 (𝑏 = 𝑌 → (𝜏𝜂))
on2ind.i ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 𝜒 ∧ ∀𝑐𝑎 𝜓 ∧ ∀𝑑𝑏 𝜃) → 𝜑))
Assertion
Ref Expression
on2ind ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑   𝜓,𝑎   𝜏,𝑎   𝑏,𝑐   𝜒,𝑏   𝑏,𝑑   𝜂,𝑏   𝑐,𝑑   𝜑,𝑐   𝜃,𝑐   𝜓,𝑑   𝑋,𝑎,𝑏   𝑌,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑑)   𝜓(𝑏,𝑐)   𝜒(𝑎,𝑐,𝑑)   𝜃(𝑎,𝑏,𝑑)   𝜏(𝑏,𝑐,𝑑)   𝜂(𝑎,𝑐,𝑑)   𝑋(𝑐,𝑑)   𝑌(𝑎,𝑐,𝑑)

Proof of Theorem on2ind
StepHypRef Expression
1 onfr 6357 . 2 E Fr On
2 epweon 7710 . . 3 E We On
3 weso 5625 . . 3 ( E We On → E Or On)
4 sopo 5565 . . 3 ( E Or On → E Po On)
52, 3, 4mp2b 10 . 2 E Po On
6 epse 5617 . 2 E Se On
7 on2ind.1 . 2 (𝑎 = 𝑐 → (𝜑𝜓))
8 on2ind.2 . 2 (𝑏 = 𝑑 → (𝜓𝜒))
9 on2ind.3 . 2 (𝑎 = 𝑐 → (𝜃𝜒))
10 on2ind.4 . 2 (𝑎 = 𝑋 → (𝜑𝜏))
11 on2ind.5 . 2 (𝑏 = 𝑌 → (𝜏𝜂))
12 predon 7721 . . . . . 6 (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎)
1312adantr 482 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎)
14 predon 7721 . . . . . . 7 (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏)
1514adantl 483 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑏) = 𝑏)
1615raleqdv 3314 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑𝑏 𝜒))
1713, 16raleqbidv 3320 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐𝑎𝑑𝑏 𝜒))
1813raleqdv 3314 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐𝑎 𝜓))
1915raleqdv 3314 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑𝑏 𝜃))
2017, 18, 193anbi123d 1437 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃) ↔ (∀𝑐𝑎𝑑𝑏 𝜒 ∧ ∀𝑐𝑎 𝜓 ∧ ∀𝑑𝑏 𝜃)))
21 on2ind.i . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 𝜒 ∧ ∀𝑐𝑎 𝜓 ∧ ∀𝑑𝑏 𝜃) → 𝜑))
2220, 21sylbid 239 . 2 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃) → 𝜑))
231, 5, 6, 1, 5, 6, 7, 8, 9, 10, 11, 22xpord2ind 8081 1 ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065   E cep 5537   Po wpo 5544   Or wor 5545   We wwe 5588  Predcpred 6253  Oncon0 6318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  naddcllem  8623  naddcom  8629
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