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Theorem pgind 49236
Description: Induction on partizan games. (Contributed by Emmett Weisz, 27-May-2024.)
Hypotheses
Ref Expression
pgind.1 (𝑥 = 𝑦 → (𝜓𝜒))
pgind.2 (𝑦 = 𝐴 → (𝜒𝜃))
pgind.3 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
Assertion
Ref Expression
pgind (𝜑 → (𝐴 ∈ Pg → 𝜃))
Distinct variable groups:   𝑦,𝐴   𝜒,𝑥   𝜓,𝑦   𝜃,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)   𝐴(𝑥)

Proof of Theorem pgind
StepHypRef Expression
1 19.8a 2181 . 2 (𝜑 → ∃𝑦𝜑)
2 19.8a 2181 . 2 (∃𝑦𝜑 → ∃𝑥𝑦𝜑)
3 nfe1 2150 . . 3 𝑥𝑥𝑦𝜑
4 nfe1 2150 . . . 4 𝑦𝑦𝜑
54nfex 2324 . . 3 𝑦𝑥𝑦𝜑
6 pgind.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
7 pgind.2 . . 3 (𝑦 = 𝐴 → (𝜒𝜃))
8 nfa1 2151 . . . 4 𝑥𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
9 nfra1 3284 . . . . . . 7 𝑦𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒
10 nfv 1914 . . . . . . 7 𝑦𝜓
119, 10nfim 1896 . . . . . 6 𝑦(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
1211nfal 2323 . . . . 5 𝑦𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
13 pgind.3 . . . . 5 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
1412, 13exlimi 2217 . . . 4 (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
158, 14exlimi 2217 . . 3 (∃𝑥𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
163, 5, 6, 7, 15pgindnf 49235 . 2 (∃𝑥𝑦𝜑 → (𝐴 ∈ Pg → 𝜃))
171, 2, 163syl 18 1 (𝜑 → (𝐴 ∈ Pg → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538   = wceq 1540  wex 1779  wcel 2108  wral 3061  cun 3949  cfv 6561  1st c1st 8012  2nd c2nd 8013  Pgcpg 49228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015  df-setrecs 49203  df-pg 49229
This theorem is referenced by: (None)
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