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Theorem pgind 47715
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 2174 . 2 (𝜑 → ∃𝑦𝜑)
2 19.8a 2174 . 2 (∃𝑦𝜑 → ∃𝑥𝑦𝜑)
3 nfe1 2147 . . 3 𝑥𝑥𝑦𝜑
4 nfe1 2147 . . . 4 𝑦𝑦𝜑
54nfex 2317 . . 3 𝑦𝑥𝑦𝜑
6 pgind.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
7 pgind.2 . . 3 (𝑦 = 𝐴 → (𝜒𝜃))
8 nfa1 2148 . . . 4 𝑥𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
9 nfra1 3281 . . . . . . 7 𝑦𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒
10 nfv 1917 . . . . . . 7 𝑦𝜓
119, 10nfim 1899 . . . . . 6 𝑦(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
1211nfal 2316 . . . . 5 𝑦𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
13 pgind.3 . . . . 5 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
1412, 13exlimi 2210 . . . 4 (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
158, 14exlimi 2210 . . 3 (∃𝑥𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
163, 5, 6, 7, 15pgindnf 47714 . 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 1539   = wceq 1541  wex 1781  wcel 2106  wral 3061  cun 3945  cfv 6540  1st c1st 7969  2nd c2nd 7970  Pgcpg 47707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-1st 7971  df-2nd 7972  df-setrecs 47682  df-pg 47708
This theorem is referenced by: (None)
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