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Theorem pgind 48009
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 2166 . 2 (𝜑 → ∃𝑦𝜑)
2 19.8a 2166 . 2 (∃𝑦𝜑 → ∃𝑥𝑦𝜑)
3 nfe1 2139 . . 3 𝑥𝑥𝑦𝜑
4 nfe1 2139 . . . 4 𝑦𝑦𝜑
54nfex 2309 . . 3 𝑦𝑥𝑦𝜑
6 pgind.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
7 pgind.2 . . 3 (𝑦 = 𝐴 → (𝜒𝜃))
8 nfa1 2140 . . . 4 𝑥𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
9 nfra1 3273 . . . . . . 7 𝑦𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒
10 nfv 1909 . . . . . . 7 𝑦𝜓
119, 10nfim 1891 . . . . . 6 𝑦(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
1211nfal 2308 . . . . 5 𝑦𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
13 pgind.3 . . . . 5 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
1412, 13exlimi 2202 . . . 4 (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
158, 14exlimi 2202 . . 3 (∃𝑥𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
163, 5, 6, 7, 15pgindnf 48008 . 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 205  wal 1531   = wceq 1533  wex 1773  wcel 2098  wral 3053  cun 3939  cfv 6534  1st c1st 7967  2nd c2nd 7968  Pgcpg 48001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2363  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fv 6542  df-1st 7969  df-2nd 7970  df-setrecs 47976  df-pg 48002
This theorem is referenced by: (None)
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