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Theorem pgind 48948
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 2179 . 2 (𝜑 → ∃𝑦𝜑)
2 19.8a 2179 . 2 (∃𝑦𝜑 → ∃𝑥𝑦𝜑)
3 nfe1 2148 . . 3 𝑥𝑥𝑦𝜑
4 nfe1 2148 . . . 4 𝑦𝑦𝜑
54nfex 2323 . . 3 𝑦𝑥𝑦𝜑
6 pgind.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
7 pgind.2 . . 3 (𝑦 = 𝐴 → (𝜒𝜃))
8 nfa1 2149 . . . 4 𝑥𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
9 nfra1 3282 . . . . . . 7 𝑦𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒
10 nfv 1912 . . . . . . 7 𝑦𝜓
119, 10nfim 1894 . . . . . 6 𝑦(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
1211nfal 2322 . . . . 5 𝑦𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
13 pgind.3 . . . . 5 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
1412, 13exlimi 2215 . . . 4 (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
158, 14exlimi 2215 . . 3 (∃𝑥𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
163, 5, 6, 7, 15pgindnf 48947 . 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 1535   = wceq 1537  wex 1776  wcel 2106  wral 3059  cun 3961  cfv 6563  1st c1st 8011  2nd c2nd 8012  Pgcpg 48940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014  df-setrecs 48915  df-pg 48941
This theorem is referenced by: (None)
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